i.^\ 


IN  MEMORIAM 
FLORIAN  CAJORl 


ELEMENTARY 


PLANE    GEOMETRY 


INDUCTIVE  AND  DEDUCTIVE 


BY 


ALFRED   BAKER,  M.A.,  F.R.S.C. 

Professor  of  Mathematics,  University  of  Toronto 


BOSTON,  U.S.A. 
GINN   &   COMPANY,  PUBLISHERS 

1903 


133 


Copyright,  1903,  by  Ginn  &  Company 


Entered  according  to  the  Act  of  Parliament  of  Canada,  in  the  Office  of  the 
Minister  of  Agriculture,  by  W.  J.  Gage  &  Co.  Limited,  Toronto,  in  the 
year  1903. 


CAJOR! 


PEEFACE 


The  geometry  of  Euclid  is  deductive.  Yet  the  processes 
of  all  sciences,  other  than  pure  mathematics,  involve  both 
induction  and  deduction.  All  the  knowledge  vv^hich  we  have 
of  life,  with  its  varied  phenomena,  is  reached  by  induction 
and  deduction.  Any  science,  then,  which  permits  the  student, 
from  a  number  of  observations,  to  reach  a  general  result, 
and  again  from  such  generalization  to  draw  conclusions,  must 
have  distinct  educational  value.  The  present  little  book  is 
an  attempt  to  make  the  processes  of  elementary  geometry 
both  inductive  and  deductive.  I  feel  that  in  making  this 
attempt  I  am  adapting  the  .study  of  Geometry  to  immature 
minds.  The  mind  of  youth  receives  its  knowledge  in  the  form 
of  isolated  facts ;  it  is  for  the  educator  to  point  out  that 
isolated  facts  fall  into  groups  and  may  be  crystallized  into 
general  conclusions.  Special  opportunities  present  themselves 
in  elementary  geometry  for  following  this  method.  Thus,  if  a 
number  of  triangles  be  accurately  constructed  with  bases  of 
45  millimetres  and  angles  at  the  bases  75°  and  62°,  by  actual 
measurement  the  learner  finds  that  all  the  sides  opposite  to 
the  angles  of  75°  are  equal,  and  likewise  those  opposite  to  the 
angles  of  62°,  and  that  the  remaining  angles  of  the  triangles  have 
the  same  magnitude.  Analogous  constructions  and  measure- 
ments being  repeated  in  a  number  of  cases,  the  learner,  as  a 
matter  of  inductive  observation,  feels  himself  justified  in 
making  the  generalization  expressed  in  the  enunciation  of 
Euclid  I.,  26.  In  the  process  the  intellectual  interest  and 
curiosity  of  the  pupil  are  excited,  and  in  reaching  the  conclu- 
sion he  feels  almost  as  if  he  had  made  the  discovery  himself. 
If,  subsequently,  geometrical  forms  are  presented  to  him  where 
he  can  utilize  his  previous  conclusion,  he  feels  with  keenness 
the  value  of  his  previous  work.  He  has,  in  fact,  been  going 
through  the  process  of  induction  and  deduction, — the  process 
through  which  every  scientific  discoverer  goes — with,  in  mini- 
ature, the  emotions  of  the  investigator. 


911324 


iv  •  Pkeface. 

It  is  claimed  that  deductive  geometry  inculcates  accuracy 
of  thought.  Most  admirably  in  this  respect  it  does  its  work. 
It  too  often  happens,  however,  that  in  the  class-room  triangles 
are  alleged  to  be  equal  which  are  ridiculously  unlike,  and 
lines  are  proved  to  be  equal  which  the  eye  tells  us  differ  in 
length  by  several  inches.  In  fact,  in  spite  of  accuracy  of 
thought,  the  utmost  contempt  for  physical  accuracy  is  often 
inculcated.  The  whole  spirit  of  the  following  pages  is  accuracy 
of  construction.  Only  by  exact  drawing  can  results  be  attained, 
and  the  pupil  will  find  that  inaccuracy  means  failure.  My 
object  is  to  mahe  the  class-room  in  geometry  a  sort  of  workshop, 
where  exactness  in  drawing  lines  of  required  length,  in  meas- 
uring lines  that  are  drawn,  in  constructing  angles  of  given 
magnitude,  in  measuring  angles  that  are  constructed,  and 
generally  in  constructing  all  figures,  is  insisted  on.  The  atti- 
tude of  the  pupil  towards  his  geometrical  figures  should,  he 
that  of  the  shilled  mechanic  towards  an  instrument  or  machine 
of  precision  which  he  is  making,  where  inaccuracy  in  measure- 
ment  would  Tnean  loss  of  time  and  of  tnaterial,  and  would  be 
considered  evidence  of  stupidity. 

I  do  not  suggest  this  book  as  a  substitute  for  the  ordinary 
works  on  deductive  geometry  used  in  the  schools,  but  rather 
as  an  introduction  to  their  study.  Hence  I  have  included 
the  leading  geometrical  facts  reached  in  such  works,  and 
have  introduced  them  in  what  is  more  or  less  an  accepted 
order.  Teachers  will  find  here  about  one  year's  work  for  a 
class  of  beginners.  If  the  pupils  pursue  the  subject  of  geom- 
etry no  further,  I  humbly  trust  that  the  practical  work  they 
have  done  in  connection  with  this  course  will  have  impressed 
the  leading  facts  of  elementary  geometry  indelibly  on  their 
minds ;  if  on  the  other  hand  they  take  up  the  study  of  deduc- 
tive geometry,  I  hope  they  will  the  better,  from  following  this 
concrete  course,  appreciate  the  absolutely  general  and  irrefra- 
gable character  of  methods  purely  deductive. 

University  of  Toeonto, 
May,  1903. 


CONTENTS. 


(At  the  close  of  Chapter  VIII.  the  suggestion  is  made  that  Chap- 
ters XIX.,  XX.,  aud  XXI.,  relating  to  similar  triaiigles,  may  at 
OJice  be  proceeded  ivitli.) 

CHAPTER  PAGE 

I.  Geometrical  Elements 9 

11.  Construction  of  Triangles 15 

^~~^III.  Equa-Lity  of  Triangles 22 

IV.  Bisection  of  Lines   and   Angles.     Perpen- 
diculars    31 

^V.  Respecting  Angles  of  a  Triangle    ...  39 

VI.  Parallel  Lines 44 

VII.  Parallelograms,  Rectangles  and  Squares.  51 
—^VIII.  Certain  Relations  in  Area  Between  Par- 
allelograms AND  Triangles 57 

__—  IX.  Squares  on  Sides  of  a  Right- Angled  Tri- 
angle    67 

^X.  The    Circle.      Its    Symmetry.      Tangents. 

Finding  of  Centre 72 

XL  Tangents  to  Circles,  and   Circles   Touch- 
ing One  Another 78 

XII.  Angles  in  a  Circle 85 

XIII.  Relation  Between  Segments  of  Intersect- 

ing Chords  ....  - 94 

XIV.  Triangles  In  and  About  Circles      ...  100 
—XV.  Circles  In  and  About  Triangles       ...  106 

XVI.  Squares  and  Circles  In  and  About  Circles 

AND  Squares Ill 

XV  n.  Regular  Polygons 117 

XV III.  Regular  Polygons  (Continued) 121 

XIX.  Similar  Triangles 127 

XX.  Similar  Triangles  {Contimied) 134 

XXI.  Similar  Triangles  (Continued) 140 


Digitized  by  the  Internet  Archive 

in  2007  with  funding  from 

IVIicrosoft  Corporation 


http://www.archive.org/details/elementaryplanegOObakerich 


INSTRUM:eNTS. 

In  the   pages   that   succeed,  the   following  instruments   are 
essential : 

1.  A  ruler  or  straigrht-edgre, 

on  which  are  marked  inches  divided  into  sixteenths,  and 
on  which  also  is  a  scale  giving  millimetres. 

This  is  used  for  drawing  straight  lines  ;  for  making 
them  of  any  required  length  ;  and  for  measuring  straight 
lines  that  are  drawn. 

2.  A  pair  of  compasses, 

one  leg  of  which  is  furnished  with  a  pencil. 

This  is  used  for  describing  circles  ;  also,  with  the  help 
of  the  ruler,  for  laying  oflf  required  distances ;  and  for 
measuring  distances  that  are  laid  off. 

3.  A  protractor. 

This  is  used  for  constructing  angles  of  any  given  num- 
ber of  degrees  ;  and  for  measuring  the  number  of  degrees 
in  any  given  angle.  It  may  also  be  used  for  determining 
whether  one  angle  is  greater  than,  equal  to,  or  less  than 
another. 

For  the  more  rapid  and  more  accurate  construction  of  figures, 
the  following  instruments  are  also  desirable  : 

4.  A  pair  of  dividers, 

both  the  legs  of  which  terminate  in  fine  points.  These 
more  accurately  than  the  compasses  will  enable  the  pupil 
to  measure  and  to  transfer  distances. 

5.  A  set-square. 

The  right  angle  has  very  frequently  to  be  constructed, 
and  its  construction  can  be  more  rapidly  effected  with  the 
set-square  than  with  the  protractor. 

7 


8  Geometky. 

6.  A  bevel. 

This  enables  us  very  rapidly  to  determine  the  equality 
or  inequality  of  angles,  and  to  construct  an  angle  equal 
to  another. 

7.  Parallel  rulers. 

While  for  drawing  lines  parallel  to  each  other  nothing 
more  is  essential  than  a  ruler  along  which  the  set- 
square  is  made  to  slide,  or  a  ruler  and  an  instrument  for 
measuring  angles,  or  a  ruler  and  compasses,  these  methods 
become  tedious  from  the  frequency  with  which  the  con- 
struction has  to  be  made.  Parallel  rulers  make  the  con- 
struction rapidly  and  accurately. 

Care  should  be  taken  to  use  a  pencil  with  a  hard  fine  point,  so 
that  lines  drawn  may  be  narrow  and  well  defined. 

Smooth  paper  will  be  found  better  than  rough. 

Points  and  the  ends  of  lines  should  be  marked  by  indentations 
made  with  a  needle  or  with  the  sharp  points  of  the  dividers. 

A  piece  of  smooth,  perfectly  flat  board,  about  a  foot  square, 
will  be  found  useful  as  a  drawing  board. 

In  all  cases  the  pupil  should  construct  for  himself  the  necessary 
figures,  and  not  content  himself  with  those  in  the  book,  which  are 
merely  intended  as  suggestions.  It  will  be  usually  found  desir- 
able to  make  figures  on  a  larger  scale  than  those  in  the  text. 


The  chapters  on  similar  triangles  may  be  taken  up,  if  thought 
desirable,  as  soon  as  the  pupil  has  obtained  an  acquaintance  with 
parallel  lines,  and  knows  that  the  opposite  sides  and  angles  of 
parallelograms  are  equal.  Prominence  may  then  be  given  to 
Exercise  17,  Chapter  xxi.,  which  suggests  a  demonstration  of  the 
47th,  Book  I.,  Euclid. 


CHAPTER  I. 

Geometrical  Elements. 
A  straight  line  :    


It  is  evidently  the  shortest  distance  between  its  ends. 


A  broken  line 


A  curved  line : 


An  angle  : 


The  size  of  the  angle  does  not  depend  on  the  lengths 
of  the  bonnding  lines  AB  and  AC,  but  on  the  amount 
of  divergence  of  these  lines  from  one  another.  Thus 
the  angle  P  is  greater  than  the  angle  Q,  and  the  angle 
R  is  less  than  the  angle  Q. 


It  is  usual  to  indicate  an  angle  by  using  one  letter, 
as  the  angle  P,  or  by  using  three  letters,  as  the  angle 
BAG.  In  the  latter  case  the  letter  at  the  angle  itself 
is  in  the  middle,  and  the  other  two  letters  lie  on  the 
arms  of  the  angle. 

9 


10 


Geometky. 


B 


If  ABC  be  a  straight  line,  and  the 
angles  DBA,  DBC  be  equal,  then  each 
of  them  is  called  a  right  angle,  and 
the  lines  DB  and  ABC  are  said  to  be 
perpendicular  to  each  other. 

Evidently  at  the  point  B  there  are  four  right  angles. 

,  B 
An  angle  which  is  less  than  a 

right  angle,  as  BAC,  is  called  an 

acute  angle. 

An  angle  which  is  greater  than 
a  right  angle,  as  EDF,  is  called 
an  obtuse  angle. 

A  circle  is  the  usual  figure  de- 
scribed on  a  flat  surface  by  means  of 
the  compasses. 

Note  the  parts  called  centre, 
radius,  and  circumference. 

All  radii  of  the  same  circle  are 
equal,  since  the  ends  of  the  compass 
legs  remain  the  same  distance  apart 
while  the  circle  is  being  described. 

A  line  through  the  centre  and  terminated  both  ways 
by  the  circumference  is  called  a  diameter,  as  CD. 

The  part  of  the  circle  on  each  side  of  a  diameter  is 
called  a  semicircle. 

A  part  of  the  circumference,  as  AB,  is  called  an  arc 
of  the  circle.  The  straight  line  joining  A  and  B  is  called 
a  chord. 

Any  line  drawn  from  a  point  without  the  circle  and 
cutting  it,  is  called  a  secant. 


Geometrical  Elements. 


11 


The  circumference  of  any  circle  is  supposed  to  be 
divided  into  360  equal  parts,  each  part  being  called  a 
degree. 


If  the  arc  AB  contains  60  degrees, 
then  the  angle  ACB  at  the  centre 
is  an  angle  of  60  degrees,  expressed 
by  60\ 

The  lines  AC,  DE,  through  the 
centre,  being  perpendicular,  each 
of  the  arcs  AD,  DC,  CE,  EA 
must  contain  90°,  and  the  angles 
ABD,  DBC,  ....  are  angles  of  90°. 

A  semicircle  contains  180°,  and 
the  straight  angle  ABC  contains 
180°. 


A  triangle: 

It  has  three  sides  and  three  angles. 

A  quadrangle : 

It  has  four  angles.    Having  four  sides, 
it  is  also  called  a  quadrilateral. 

A  straight  line  joining  two  opposite  corners  of  a 
quadrilateral  is  called  a  diagonal. 

Figures  contained  by  more  than  funr  straight  lines 
are  called  polygons. 

A  straight  line  has  evidently  throughout  its  entire 
length  the  same  direction. 


12 


Geometry. 


Two  straight  lines  which  have 
the  same  direction  are  said  to  be 
parallel  to  one  another. 


Parallel  straight  lines  cannot  intersect.  For  if  they 
did,  at  the  point  of  intersection  they 
would  have  different  directions,  and 
would  therefore  have  different  direc- 
tions throughout  their  entire  lengths, 
and  hence  would  not  be  parallel. 

To  construct  with  the  protractor 
at  the  point  A  in  the  line  AB  an 
angle  of  any  required  magnitude, 
say  63° :  Place  the  centre  of  the 
protractor  at  A,   and  let   the   line         ^  ^ 

joining  the  centre  with  the  point  on  the  circumference 
which  indicates  0°,  rest  along  AB.  At  the  point  where 
the  63°  line  meets  the  circumference  make  a  fine  mark, 
C,  on  the  paper.  Removing  the  protractor,  join  AC. 
The  angle  BAG  is  of  magnitude  63°. 


Exercises. 

All  figures  In  tills  and  sncceeding  exercises  mnst  be  accurately 
constructed  >vitli  instruments. 

1.  With  the  dividers  (or  compasses)  take  off  on  the  ruler  distances 
8,  11,  17,  34  ...  .  millimetres.  With  the  points  of  the  dividers 
mark  on  your  paper  points  at  these  distances  from  each  other.  With 
the  ruler  draw  straight  lines  joining  each  pair  of  points,  thus  getting 
straight  lines  of  lengths  8,  11,  17,  34  .  .  .  millimetres. 

2.  With  the  compasses  describe  circles  having  radii  of  lengths  5, 
7,  10,  .  .   .  sixteenths  of  an  inch. 

3.  With  the  protractor  construct  angles  of  magnitude  10°,  15°,  25°, 
30°,  37°,  43°, 


EXEKCISES.  13 

4.  With  the  bevel  construct  a  second  set  of  angles  of  the  foregoing 
magnitudes,  using  these  angles  to  set  the  bevel. 

5.  Draw  five  straight  lines  of  different  lengths,  and  with  the  dividers 
and  rule  measure  their  lengths  in  inches  and  sixteenths  of  an  inch. 
Measure  also  their  lengths  in  millimetres. 

6.  Construct  five  angles,  and,  using  the  bevel,  determine  which  is 
greatest  and  which  least.  Arrange  them  in  order  of  magnitude. 
Using  the  protractor,  measure  their  magnitude  to  the  nearest  degree. 

7.  Draw  five  straight  lines  of  different  lengths,  and  with  the  eye 
endeavor  to  judge  their  lengths  (1)  in  inches  and  fractions  of  an 
inch,  (2)  in  millimetres.  Afterwards  test  the  correctness  of  your 
judgment  by  actually  measuring  the  lines. 

8.  Construct  five  angles  of  different  magnitudes,  and  with  the  eye 
endeavor  to  judge  the  number  of  degrees  in  each.  Afterwards  test 
the  correctness  of  your  judgment  by  actually  measuring  the  angles 
with  the  protractor. 

9.  With  the  eye  endeavor  to  judge  the  lengths  or  heights  of  various 
objects  in  the  room,  at  a  distance  from  you.  Afterwards  test  the 
correctness  of  your  judgment  by  actually  measuring  the  lengths  or 
heights. 

10.  A  and  B  being  two  distant  objects  and  your  eye  being  at  C, 
endeavor  with  the  eye  to  judge  the  angle  which  these  objects  sub- 
tend at  your  eye,  i.e.,  the  angle  ACB.  Afterwards  sight  the  inside 
edges  of  the  legs  of  the  bevel  towards  A  and  B,  and  then  placing  the 
bevel  on  the  protractor,  roughly  measure  in  this  way  the  angle  ACB, 
so  correcting,  if  necessary,  your  judgment. 

11.  Draw  any  two  lines  of  different  lengths,  and  draw  a  line  equal 
to  their  difference. 

12.  Draw  any  line,  and  draw  another  line  three  times  as  long  as 
the  former. 

13.  Construct  two  angles  of  different  magnitudes,  and  with  the 
bevel  constructing  two  adjacent  angles  equal  to  them,  form  an  angle 
equal  to  their  difference.  Measure  with  the  protractor  the  number  of 
degrees  in  the  original  angles  and  in  the  difference,  and  compare. 

14.  Construct  two  angles  of  different  magnitudes,  and  with  the 
bevel  constructing  two  adjacent  angles  equal  to  them,  form  an  angle 
equal  to  their  sum.  Measure  with  the  protractor  the  number  of 
degrees  in  the  original  angles  and  in  the  sum,  and  compare. 


14  G-EOMETRY. 

15.  Construct  an  angle  of  30°.  With  the  bevel  construct  two  other 
angles  equal  to  it,  one  on  each  side  of  the  first,  the  three  bounding 
lines  radiating  from  the  same  point.  What  positions  do  the  outside 
lines  of  your  figure  occupy  with  respect  to  each  other,  and  why  ? 
Test  with  an  instrument. 

16.  Construct  an  angle  of  60°.  With  the  bevel  construct  five  other 
angles  equal  to  it,  each  adjacent  to  the  preceding,  the  bounding  lines 
all  radiating  from  the  same  point.  What  positions  do  the  first  and 
last  lines  of  these  angles  occupy  with  respect  to  each  other,  and  why  ? 

17.  In  the  figure  of  the  preceding  question,  if  0  be  the  point  from 
which  the  lines  radiate,  measure  ofi"  with  the  dividers  on  these  lines 
equal  lengths,  OA,  OB,  OC,  OD,  OE,  OF.  What  do  you  observe  as 
to  the  lengths  AB,  BC,  CD,  DE,  EF,  FA  ? 

18.  Fold  a  piece  of  paper  so  as  to  get  a  straight  crease.  Fold  the 
crease  over  on  itself.  How  many  degrees  in  each  of  the  four  angles 
so  obtained,  and  why  ? 

19.  With  a  needle  mark  two  points.  Join  them,  using  ruler  and  a 
fine  pencil.  Turn  the  ruler  over  to  the  other  side  of  the  two  points 
and  again  join  them.  What  quality  in  the  ruler  may  you  test  in  this 
way? 

20.  At  points  on  your  paper  some  distance  from  one  another,  con- 
struct two  angles,  as  nearly  as  you  can  judge,  equal.  Test  with  an 
instrument  the  correctness  of  your  judgment. 

21.  Through  what  angle  does  the  minute-hand  of  a  clock  move  in 
20  minutes  ?  Through  what  angle  does  the  hour-hand  move  in  the 
same  time  ? 

22.  Describe  a  circle,  and,  supposing  it  intended  for  the  face  of  a 
clock,  mark  the  points  where  the  usual  Roman  numerals  should  be 
placed. 

23.  One  side  of  a  piece  of  paper  being  a  straight  line,  tear  the 
remaining  boundary  into  any  irregular  shape.  With  your  protractor 
convert  this  paper  into  a  protractor,  so  as  to  mark  angles  at  intervals 
of  10°,  the  markings  being  on  the  irregular  edge  of  the  paper. 


CHAPTER  II. 

Construction  of  Triangles. 

1.  Take  a  line  AB  of  any  length. 
First  with  A  as  centre,  then  with  B 
as  centre,  and  in  both  cases  with  the 
same  radius  AB,  describe  portions  of 
circles  so  that  they  intersect,  as  indi- 
cated, at  C.  Then  the  three  lines 
AB,  BC,  CA  are  all  equal.  The  tri- 
angle CAB,  which  has  thus  all  its  sides  equal,  is  called 
an  equilateral  triangle. 

Adjust  the  bevel  to  each  of  the  angles  of  this  triangle, 
and  compare  their  magnitudes. 

Construct  equilateral  triangles  whose  sides  are  14,  21, 
30,  40  .  .  .  sixteenths  of  an  inch. 

Apply  the  bevel  to  all  the  angles  of  these  triangles, 
and  compare  their  magnitudes. 

Cut  accurately  any  one  of  these  equilateral  triangles 
from  the  paper,  and,  clipping  off  the  angles,  fit  them 
on  one  another,  and  on  the  angles  of  the  other  equi- 
lateral triangles,  so  as  to  compare  their  magnitudes. 

The  result  of  our  observations  is  that  the  angles  in 
an  equilateral  triangle  are  equal  to  one  another, 
and  are  equal  to  the  angles  in  any  other  equi- 
lateral triangle. 

Using  the  bevel,  construct  three 
angles  adjacent  to  one  another,  in 
the  way  indicated  in  the  annexed 
figure,  each  angle  being  equal  to 
the  angle  of    an    equilateral    tri-  '^ 

Note.— It  is  well  to  mark  ou  lines  and  angles  their  magnitudes,  when 
known. 

15 


16 


Geometey. 


angle.  Applying  the  ruler,  it  will  be  found  that  CA 
and  AB  are  in  the  same  straight  line.  Hence  it  appears 
that  the  three  angles  of  any  equilateral  triangle  are 
together  equal  to  180°,  and  any  one  of  the  angles  in 
such  a  triangle  is  60°. 

Measure  the  angles  in  several  of  the  equilateral  tri- 
angles with  the  protractor  to  verify  this. 

2.  Take  a  line  AB  of,  say,  25 
millimetres  in  length,  and  with 
centres  A  and  B  describe  portions 
of  circles  intersecting  as  indicated 
at  C,  each  circle  having  the  same 
radius,  say  35  millimetres.  Draw 
lines  from  C  to  A  and  B.  Then 
the  triangle  CAB  has  two  sides 
equal.  A  triangle  with  two  of  its 
sides  equal  is  called  an  isosceles 
triangle. 

Adjusting  the  bevel  to  the  angles  CAB  and  CBA,  com- 
pare their  magnitudes. 

Compare  also  the  sizes  of  these  angles  by  accurately 
cutting  the  triangle  out  of  the  paper,  and  placing  the 
triangle  reversed  in  the  vacant  space  left  in  the  paper, 
so  that  the  angle  B  rests  in  the  space  A. 

Compare  also  the  sizes  of  these  angles  by  folding 
the  triangle  along  the  line  from  C  to  the  middle  of  AB. 

Construct  the  following  isosceles  triangles : 
Base  1  in.,  each  side  2  in. 
Base  3  in.,  each  side  2  in. 
Base  2 J  in.,  each  side  2||  in. 


Construction  of  Triangles.  17 

In  each  case  compare  the  magnitudes  of  the  angles  at 
the  base. 

The  result  of  our  observations  is  that  the  angles 
at  the  base  of  an  isosceles  triangle  are  equal. 

Of  course  it  would  follow  from  this  that  all  the  angles 
in  an  equilateral  triangle  are  equal,  as  we  have  already 
seen. 

Prolong  the  sides  CA,  CB,  and 
adjusting  the  bevel  to  the  angles 
BAD,  ABE,  on  the  other  side  of 
the  base,  they  will  be  found  to  be 
equal.  This  may  also  be  reasoned 
out  as  follows :  The  angles  on  one 
side  of  a  straight  line  at  any  point 
in  it  make  up  180°.  But  the 
angles  CAB   and  CBA    are  equal.    '"  ^E- 

Therefore  the  remaining  angles  BAD  and  ABE  are  also 
equal. 

3.  Taking  any  line  AB,  with  the 
bevel  or  protractor  construct  equal 
angles  at  A  and  B,  and  produce  the 
bounding  lines  of  these  angles  to 
meet  in  C.  Then  employing  the 
dividers  or  compasses,  compare  the 
lengths  of  the  sides  CA,  CB,  of  the 
triangle  CAB. 

Construct  the  following  triangles: 

Base  25  millimetres,  each  of  angles  at  base  75'. 
Base  70  millimetres,  each  of  angles  at  base  SO". 
Base  3  in.,  each  of  angles  at  base  45°. 


18 


Geometry. 


In  each  case  compare  the  magnitudes  of  the  sides 
adjacent  to  the  equal  angles. 

The  result  of  our  observations  is  that  if  two  angles 
of  a  triangle  are  equal,  the  sides  opposite  to 
these  angles  are  also  equal. 

In  the  case  of  each  of  the  above  triangles  measure 
the  size  of  the  angle  at  the  top,  or  vertex,  of  the  tri- 
angle, and  find  the  total  number  of  degrees  in  the  three 
angles  of  each  triangle. 

4.  Take  a  line  AB  of 
length  35  millimetres,  and 
with  centres  A  and  B,  and 
radii  45  and  50  milhmetres 
respectively,  describe  por- 
tions of  circles,  so  that 
they  intersect  at  C.  Join 
CA  and  CB.  We  have  thus 
a  triangle  CAB  whose  sides 
are  unequal,  called  a  sca- 
lene triangle. 

Construct  the  following 
triangles : 

With  sides  8,  5  and  6  inches. 

With  sides  70,  80  and  100  millimetres. 

With  sides  3J,  4J  and  2J  inches. 


With  the  bevel  lay  off  three 
angles  adjacent  to  one  another, 
equal  to  the  angles  of  each  tri- 
angle, in  the  way  indicated  in  the 

adjacent   figure;    and    determine      

the  positions   of   the   initial   and     ^  ^ 

final  lines,  LM,  LK,  with  respect  to  one  another. 


M 


Exercises.  19 

What  conclusion  do  you  draw  as  to  the  total  number 
of  degrees  in  the  three  angles  of  each  of  these  triangles! 

Can  you  construct  a  triangle  with  sides  of  30,  50  and 
90  millimetres,  or  with  sides  of  2,  3  and  6  inches? 
Attempt  the  construction. 

What  relation  must  exist  between  the  given  lengths, 
that  a  triangle  may  be  constructed  with  sides  of  such 
lengths  1 

Exercises. 

Teacbers  are  advised  to  have  their  classes  work  l>ut  a  few  of  tlie 
exercises  nt  the  close  of  each  chapter.  The  time  of  pupils  should 
be  chiefly  occupied  iu  verifying  the  geometric  truths  reached  lu 
the  text. 

1.  At  a  given  point  in  a  straight  line  construct  an  angle  of  60°, 
using  only  compasses  and  ruler. 

2.  Construct  an  isosceles  triangle,  and  produce  the  base  both  ways. 
What  do  you  note  as  to  the  magnitudes  of  the  exterior  angles  so 
formed  ? 

3.  Construct  a  triangle  with  sides  30,  50,  70  millimetres.  With 
the  bevel  or  protractor  determine  which  is  the  greatest  angle  and 
which  the  least. 

4.  The  angle  at  the  vertex  of  an  isosceles  triangle  is  75°,  and  each 
of  the  equal  sides  is  2  inches.     Construct  the  triangle. 

5.  At  A  in  the  line  AB  construct  the  angle  BAD  of  40°,  and  at  B 
the  angle  ABC  of  120°.  Produce  AD,  BC  to  meet.  Measure  the  size 
of  the  third  angle  of  this  triangle.  Which  is  the  greatest  side  and 
which  the  least  ? 

6.  On  one  side  of  BC  describe  an  equilateral  triangle  ABC,  and  on 
the  other  side  of  BC  describe  an  isosceles  triangle  DBC.  Join  AD. 
Take  a  number  of  points  E,  F,  G,  .  .  .  in  AD.  What  do  you  note  as 
to  the  lengths  of  EB  and  EC  ;  of  FB  and  FC  ;  of  GB  and  GC,  .  .  .  ? 

7.  Make  the  same  construction  as  in  the  preceding  question,  but 
with  the  isosceles  triangle  on  the  same  side  of  BC  as  the  equilateral. 
Produce  AD  both  ways.  What  again  do  you  note  as  to  the  distances 
of  any  point  in  AD,  or  AD  produced,  from  B  and  C  ? 


20  Geometky. 

8.  On  BC  describe  an  equilateral  triangle  ABC,  and  on  the  other 
side  of  BC  describe  a  scalene  triangle  DBC.  Join  AD.  Take  a  num- 
ber of  points  E,  F,  G,  .  .  .  in  AD.  What  do  you  note  as  to  the 
lengths  of  EB  and  EC  ;  of  FB  and  FC  ;  of  GB  and  GC, ? 

9.  Repeating  the  figure  of  6,  take  in  BC,  and  on  the  same  side  of 
AD,  a  number  of  points  K,  L,  M,  N.  .  .  .  What  do  you  note  as  to 
the  lengths  of  AK,  AL,  AM,  AN,  .  .  .  ?  Do  they  seem  to  follow 
any  law  as  to  magnitude  ? 

10.  Describe  an  equilateral  triangle  ABC.  On  BC  describe  an 
equilateral  triangle  DBC  ;  on  CA  an  equilateral  triangle  ECA  ;  and 
on  AB  an  equilateral  triangle  FAB.  Join  AD,  BE,  CF.  What  do 
you  observe  as  to  the  positions  of  the  lines  DC,  CE  with  respect  to 
one  another  ;  of  EA,  AF  ;  and  of  FB,  BD  ? 

11.  In  the  preceding  question  mark  all  the  angles  that  are  equal  to 
one  another  ;  also  all  the  lines  that  are  equal  to  one  another. 

What  triangles  are  isosceles? 

Do  you  observe  any  equilateral  four-sided  figures  ? 

How  many  equilateral  triangles  are  there  ? 

12.  With  centre  A,  outside  a  straight  line,  describe  a  circle  of  such 
radius  as  to  cut  the  line  in  two  points,  B  and  C.  What  sort  of  tri- 
angle is  ABC  ? 

13.  In  the  figure  of  the  preceding  question  find  on  the  side  of  BC 
remote  from  A,  a  point  D,  such  that  a  circle  with  D  as  centre  can  be 
described  to  pass  through  both  B  and  C. 

14.  B  and  C  being  two  points  in  a  line,  find  on  either  side  of  the 
line  points  K,  L,  M,  N,  .  .  .  such  that  a  circle  may  be  described, 
with  any  one  of  them  as  centre,  to  pass  through  B  and  C.  What  do 
you  observe  as  to  the  positions  of  K,  L,  M,  N,  .  .  .  with  respect  to 
one  another? 

15.  Construct  a  scalene  triangle  ABC,  and  on  the  side  of  BC  away 
from  A,  describe  a  triangle  DBC,  with  DB  =  AB,  and  DC  =  AC.  Join 
AD.  What  triangles  in  the  figure  are  isosceles?  What  inference  can 
you  draw  as  to  the  angles  BAC,  BDC  ?  Is  any  line  in  the  figure 
bisected?  What  are  the  angles  at  the  intersection  of  BC  and  AD? 
(Apply  set-square.) 


Exercises.  21 

16.  Construct  a  scalene  triangle  ABC,  and  on  the  side  of  BC 
remote  from  A,  describe  a  triangle  DBC  with  DB  =  AC,  and  DC=  AB. 
Join  AD.     How  do  AD  and  BC  appear  to  divide  each  other  ? 

Repeat  the  construction  several  times  with  different  pairs  of  tri- 
angles, and  note  whether  the  same  peculiarity  of  division  occurs 
in  each  case. 

17.  Construct  a  scalene  triangle  ABC.  On  the  other  side  of  BC 
construct  DBC  with  DB  =  AB,  and  DC  =  AC  ;  on  the  other  side  of  AC 
construct  EAC  withCE  =  CB,  and  AE=AB  ;  on  the  other  side  of  AB 
construct  FAB  with  BF  =  BC,  and  AF  =  AC.  Join  AD,  BE,  CF. 
What  lines  in  the  figure  are  bisected  ?  What  triangles  are  isosceles? 
What  angles  are  right  angles  ?  How  many  right-angled  triangles  are 
there  ? 

18.  Construct  a  triangle  ABC  (BC=:47,  CA  =  40,  AB  =  27  milli- 
metres). On  the  other  side  of  BC  construct  DBC  with  DB  =  AC, 
and  DC  =  AB  ;  on  the  other  side  of  AC  construct  EAC  with  EC  =  AB, 
and  EA  =  BC  ;  on  the  other  side  of  AB  construct  FAB  with  FA  =  BC, 
andFB  =  AC.  Join  AD,  BE,  CF.  What  are  the  positions  of  DC, 
and  CE  with  respect  to  each  other  ;  also  EA,  AF  ;  and  FB,  BD? 

What  lines  in  the  figure  are  bisected  ?  Has  any  line  the  third 
part  cut  off?     Has  any  line  the  sixth  ? 

19.  If  two  sides  of  a  triangle  are  unequal,  the  angles  opposite  to 
them  are  unequal.  (Suppose  the  angles  equal  and  prove  an  ab- 
surdity.) 

20.  If  two  angles  of  a  triangle  are  unequal,  the  sides  opposite  to 
them  are  unequal. 


CHAPTER  III. 


Equality  of  Triangrles. 

1.  Construct  two  triangles,  each  with  sides  of  lengths 
IJ,  IJ  and  IJ  inches,  as  indicated  in  the  adjacent 
figures. 


Adjust  the  bevel,  or  the  protractor,  to  the  angle  A, 
and  also  to  the  angle  D,  and  carefully  compare  the 
magnitudes  of  these  angles.  In  like  manner  compare 
the  magnitudes  of  the  angles  B  and  E,  and  also  the 
magnitudes  of  the  angles  C  and  F. 

Next  cut  both  triangles  from  the  paper,  and  place 
one  triangle  upon  the  other  so  that  the  corresponding 
angular  points  coincide.  From  this  superposition  what 
conclusion  do  you  draw  as  to  the  areas  of  the  triangles  ? 

Repeat  the  same  construction,  measurement,  and 
superposition  with  two  triangles  whose  sides  are  4,  2 
and  4 J  inches;  with  two  whose  sides  are  50,  80  and 
100  millimetres  J  etc. 

22 


Equality  op  Triangles. 


23 


The  result  of  our  observations  in  these  cases  is  that 
if  two  triangles  have  their  sides  equal,  the 
angles  which  are  opposite  to  equal  sides  are 
equal,  and  the  areas  are  equal.  In  other  words 
two  such  triangles  are  the  same  triangle  in  different 
positions. 

Another  way  of  stating  the  fact  is  to  say  that  if 
the  sides  of  a  triangle  are  fixed,  the  angles  are  fixed, 
and  the  area  is  fixed. 

2.  Construct  two  angles,  BAG  and  EDF,  each  of  30°. 
On  sides  of  these  angles  measure  off  distances  AB  and 
DE,  each  of  length  40  millimetres;  and  also  distances 
AC  and  DF,  each  of  length  51  millimetres.  Join  BC 
and  EF,  thus  forming  two  triangles,  ABC  and  DEF. 

A  D 


Adjust  the  bevel,  or  protractor,  to  the  angle  B,  and 
also  to  the  angle  E,  and  carefully  compare  the  magni- 
tudes of  these  two  angles.  In  like  manner  compare  the 
magnitudes  of  the  angles  C  and  F.  With  the  dividers 
compare    the    magnitudes    of    the    sides   BC   and   EF. 


24  G-EOMETKY. 

Further,  cut  one  triangle  from  the  paper,  and  place  it 
upon  the  other.  From  this  superposition  what  conclu- 
sion do  you  draw  as  to  the  areas  of  the  two  triangles? 

Repeat  the  same  construction,  measurement,  and 
superposition  with  the  following  triangles: 

Two  whose  sides  are  If  and  2J  inches,  and  included 

angle  30°. 
Two  whose  sides   are   30   and   110    millimetres,    and 

included  angle  78°. 
Two  whose  sides   are  IJ  and  2  inches,  and  included 

angle  135°. 

The  result  of  our  observations  in  all  these  cases  is 
that  if  two  triangles  have  two  sides  in  each 
equal,  and  the  angles  included  by  these  two 
sides  equal,  then  the  remaining  sides  are  equal, 
and  the  angles  opposite  to  equal  sides  are 
equal,  and  the  triangles  are  equal  in  area. 
In  other  words  two  such  triangles  are  the  same  triangle 
in  different  positions. 

Another  way  of  stating  the  fact  is  to  say  that  if 
two  sides  and  the  included  angle  of  a  triangle  are  fixed, 
the  remaining  side  and  angles  are  fixed,  and  the  area 
is  fixed. 

3.  In  the  case  of  aU  the  triangles 
in  1  and  2,  lay  off,  with  the  bevel, 
three  angles  adjacent  to  one  an- 
other, equal  to  the  three  angles  of 
each  triangle,  in  the  way  indicated 
in  the  figure.  Determine  the  posi- 
tions of  the  initial  and  final  lines,  LM  and  LK,  with 
respect  to  one  another. 


Equality  of  Triangles.  25 

The  result  of  such  an  examination  will  be  found  to 
be  that  the  lines  KL  and  LM  are  in  the  same  straight 
line,  i.e.,  the  -sum  of  the  three  angles  in  each  of 
these  triangles  is  two  right  angles,  or  180°. 

4.  It  is  proposed  to  show  that  the  sum  of  the  three 
angles  of  any  triangle  must  be  two  right  angles,  or 
180°: 

Construct  a  triangle 
ABC,  and  place  a  pencil 
in  the  position  DC. 
Turn  the  pencil  through 
the  angle  BCA,  in  the 
direction  indicated  by 
the  arrow  head,  to  the 
position  EC.  Slide  it 
along  CA,  towards  A,  to 
the  position  FG,  and  turn  it  through  the  angle  CAB, 
to  the  position  HK.  Slide  it  along  AB  to  the  position 
BL,  and  turn  it  through  the  angle  B,  to  the  position 
BM. 

The  pencil  has  rotated  through  all  the  angles  of  the 
triangle.  But  in  its  final  position  BM  it  points  in  a 
direction  just  opposite  to  its  first  position  DC,  and 
therefore  must  have  rotated  through  180°.  Hence  all 
the  angles  of  this  (which  is  any)  triangle  must 
together  equal  180  ,  or  two  right  angles. 

It  foUows  that  if  two  triangles  have  two 
angles  in  the  one  equal  to  two  angles  in  the 
other,  the  third  angle  in  one  triangle  is  equal 
to  the  third  angle  in  the  other. 


26  GrEOMETEY. 

5.  Construct  two  triangles,  ABC,  DEF,  each  with  base 
1|  inch,  and  angles  at  the  base  79°  and  57°. 


It  follows,  from  4,  that  the  remaining  angles  at  A 
and  D  are  equal,  each  being  44°.  Putting  the  points 
of  the  dividers  on  A  and  B,  and  carrying  the  dividers, 
so  adjusted,  to  DE,  compare  the  magnitudes  of  AB  and 
DE.  In  like  manner  compare  the  magnitudes  of  AC 
and  DF. 

Next,  cutting  one  of  the  triangles  from  the  paper, 
place  it  upon  the  other.  From  this  superposition  what 
conclusion  do  you  draw  as  to  the  areas  of  the  triangles  ? 

Kepeat  the  same  construction,  measurement,  and 
superposition  with  the  following  triangles: 

Two  whose  bases  are  If  in.,  and  angles  adjacent  to 
base  38°  and  110°. 

Two    whose   bases    are    90    millimetres,    and    angles 
adjacent  to  base  89°  and  57°. 

Two  whose  bases  are  3^  in.,  and  angles  adjacent  to 
the  base  49°  and  95°. 


EXEBCISES.  27 

The  result  of  our  observations  in  all  these  cases  is 
that  if  two  triangles  have  their  bases  equal, 
and  angles  adjacent  to  the  bases  equal,  the 
remaining  angles  are  equal,  and  the  sides  op- 
posite to  equal  angles  are  equal,  and  the 
areas  are  equal.  In  other  words  they  are  the  same 
triangle  in  different  positions. 

Another  way  of  stating  the  fact  is  to  say  that  if  a 
side  of  a  triangle  and  the  angles  adjacent  to  this  side 
are  fixed,  then  the  remaining  angle  and  sides  are  fixed, 
and  area  is  fixed. 

6.  The  following  fact,  demonstrated  in  Chapter  VI., 
may  be  of  service  in  connection  with  the  succeeding 
exercises  : 

The  vertically  opposite 
angles  AEC  and  BED  are 
equal  J  and  also  the  vertically 
opposite  angles  AED  and  BEC. 


Exercises. 

lu  numerical  exercises,  such  as  tlie  first  twelve,  the  teacher  should 
solve  the  triaugles  by  the  usual  trigonometrical  formulte,  that  he 
may  inform  the  class  as  to  the  closeness  of  their  approximations 
reached  by  instrumental  methods. 

1.  The  sides  of  a  triangle  are  35,  52  and  63  millimetres.  Con- 
struct the  triangle ;  and  with  the  protractor  measure  the  angles  to 
the  nearest  degree. 

2.  The  sides  of  a  triangle  are  36,  48  and  60  millimetres.  Con- 
struct the  triangle ;  and  with  the  protractor  measure  the  angles  to 
the  nearest  degree. 

3.  The  sides  of  a  triangle  are  66,  90  and  31  millimetres.  Con- 
struct the  triangle  ;  and  measure  the  angles  to  the  nearest  degree. 


28  Geometky. 

4.  Two  sides  of  a  triangle  are  2^  and  1^  inches,  and  the  included 
angle  is  47°.  Construct  the  triangle  ;  and  measure  the  remaining 
side  to  the  nearest  sixteenth  of  an  inch,  and  the  remaining  angles  to 
the  nearest  degree. 

5.  Two  sides  of  a  triangle  are  50  and  68  millimetres,  and  the  in- 
cluded angle  is  94°.  Construct  the  triangle  ;  and  measure  the  re- 
maining side  to  the  nearest  millimetre,  and  the  remaining  angles  to 
the  nearest  degree. 

6.  Tw-o  sides  of  a  triangle  are  5^  and  6^  inches,  and  the  included 
angle  is  54°.  Construct  the  triangle  ;  and  measure  the  remaining 
side  to  the  nearest  sixteenth  of  an  inch,  and  the  remaining  angles  to 
the  nearest  degree. 

7.  Two  angles  of  a  triangle  are  55°  and  65°,  and  the  side  adjacent 
to  them  is  27  millimetres.  Construct  the  triangle  ;  and  measure  the 
remaining  angle  to  the  nearest  "degree,  and  the  remaining  sides  to 
the  nearest  millimetre. 

8.  Two  angles  of  a  triangle  are  107°  and  27°,  and  the  side  adjacent 
to  them  is  50  millimetres.  Construct  the  triangle  ;  and  measure  the 
remaining  angle  to  the  nearest  degree,  and  the  remaining  sides  to  the 
nearest  millimetre. 

9.  Two  angles  of  a  triangle  are  53°  and  66°,  and  the  side  adjacent 
to  them  is  4  inches.  Construct  the  triangle  ;  and  measure  the  re- 
maining angle  to  the  nearest  degree,  and  the  remaining  sides  to  the 
nearest  sixteenth  of  an  inch. 

10.  The  sides  of  a  triangle  are  4,  6  and  7  inches.  Construct  the 
triangle  ;  and  measure  the  angles  to  the  nearest  degree. 

11.  Two  sides  of  a  triangle  are  90  and  70  millimetres,  and  the  in- 
cluded angle  is  58°.  Construct  the  triangle ;  and  measure  the  re- 
maining side  to  the  nearest  millimetre,  and  the  remaining  angles  to 
the  nearest  degree. 

12.  Two  angles  of  a  triangle  are  30°  and  128°,  and  the  side  ad- 
jacent to  them  is  2^  inches.  Construct  the  triangle  ;  and  measure 
the  remaining  angle  to  the  nearest  degree,  and  the  remaining  sides 
to  the  nearest  sixteenth  of  an  inch. 

13.  Two  lines  AB  and  CD  intersect  in  E,  and,  with  the  dividers, 
AE  and  EB  are  taken  equal  to  one  another,  and  also  CE  and  ED 
equal  to  one  another,  Join  AC,  CB,  BD,  DA.  What  lines,  angles 
and  triangles  are  equal  to  one  another  ?     Give  proof. 


EXEKCISES.  29 

14.  A  triangle  ABC  is  described,  and  on  the  other  side  of  BC  the 
triangle  DBC  is  constructed  with  DB  =  AB  and  DC=AC.  AD  is 
joined.  What  lines,  angles  and  triangles  are  equal  to  one  another  ? 
Give  proof.     What  angles  are  right  angles  ? 

15.  A  triangle  ABC  is  described,  and  on  the  other  side  of  BC  the 
triangle  DBC  is  constructed  with  DB  =  AC  and  DC=AB.  AD  is 
joined.  What  lines,  angles  and  triangles  are  equal  to  one  another? 
Give  proof. 

16.  A  triangle  ABC  is  described,  and  on  the  same  side  of  BC 
another  triangle  DBC  is  described  with  DB=AC  and  DC  =  AB. 
AD  is  joined.  What  lines,  angles  and  triangles  in  the  figure  are 
equal  ?     Give  proof. 

If  BA    and  CD    be    produced    to    meet  in   E,  what  are   the 
triangles  EAD,  EBC  ?     Give  reasons. 

17.  From  two  lines  diverging  from  A,  equal  lengths  AB,  AC  are 
cut  off,  and  also  equal  lengths  AD,  AE.  CD,  BE,  BC  and  DE  are 
joined.  What  lines,  angles  and  triangles  in  the  figure  are  equal? 
Give  proof. 

18.  Two  circles  have  the  same  centre  O.  AOB  is  a  diameter  of 
one,  and  COD  a  diameter  of  the  other.  AC  and  BD  are  joined. 
What  lines,  angles  and  triangles  in  the  figure  are  equal  ? 

19.  Equal  lines  AB,  AC  are  drawn,  making  equal  angles  with  AE 
on  opposite  sides  of  it.  At  B  and  C  equal  angles  ABF,  ACG  are 
constructed  towards  the  same  side.  If  AE,  BF  and  CG  be  produced, 
will  they  hit  the  same  point  ?     Give  proof. 

20.  Describe  ABC,  DBC,  two  isosceles  triangles  on  the  same  base 
BC,  but  on  opposite  sides  of  it.  How  does  AD  divide  the  angles 
BAC,  BDC  ?     Give  proof. 

21.  With  centres  A  and  B  two  circles  are  described,  intersecting  at 
C  and  D.  How  are  the  angles  CAD,  CBD  divided  by  AB  ?  How  is 
CD  divided  by  AB  ?  What  are  the  angles  at  the  intersection  of  AB 
and  CD  ?     Give  proof. 

22.  Construct  an  equilateral  triangle  ABC.  At  B  and  C  cjonstruct 
equal  angles  GBC,  GCB.  Join  AG.  How  does  AG  divide  the  angle 
BAC?    Give  proof. 


30  GrEOMETRY. 

23.  With  O  as  centre  describe  a  circle,  and,  with  the  dividers,  take 
three  points  on  the  circumference.  A,  B,  C,  such  that  the  chords  AB, 
BC  are  equal.  How  does  OB  divide  the  angles  ABC,  AOC  ?  How 
does  OB  divide  AC,  and  what  are  the  angles  at  the  point  of  intersec- 
tion ?    Give  proof. 

24.  ABC  is  any  triangle.  The  side  BC  is  produced  to  D,  CA 
to  E,  and  AB  to  F.  How  many  degrees  are  there  in  the  sum  of  the 
angles  ACD,  BAE,  CBE  ?    Verify  by  measurement  and  addition. 


CHAPTBR  IV. 


Bisection  of  Lines  and  Angles.    Perpendiculars. 

1.  To  bisect  a  straight  line. 

Suppose  AB  the  line  to  be  bisected.  With  A  and  B  as 
centres  describe  portions  of  circles  with  equal  radii 
intersecting  at  C,  and  with  the  same  centres  describe 
portions  of  circles  with  equal  radii,  intersecting  at  D. 
Then  if  CD  be  drawn,  it  bisects  AB  at  right  angles. 

For,  using  the  dividers,  it  will  be 
found  that  AE  and  EB  are  equal ;  and, 
using  the  protractor  or  set-square,  all 
the  angles  at  E  will  be  found  to  be 
90°. 

Or  again,  we  would  conclude  that 
AE  and  EB  are  equal,  and  that  the 
angles  at  E  are  right  angles,  from  the 
symmetry  of  the  figure  with  respect 
to  the  line  CD — the  figure  on  one  side  of  this  line  being 
just  the  same  as  the  figure  on  the  other  side,  but  turned 
in  the  opposite  direction. 

Or  again,  we  may  '^ reason  out"  the  equality  of  AE 
and  EB,  and  that  the  angles  at  E  are  right  angles, 
as  follows:  Since  the  triangles  ACD,  BCD  have  their 
sides  equal,  they  are  equal  in  all  respects  (Ch.  III.,  1). 
Hence  the  angles  at  C  are  equal  j  also  the  sides  about 
these  angles,  AC,  CE,  and  BC,  CE,  are  equal  5  therefore 
(Ch.  III.,  2)  the  triangles  ACE  and  BCE  are  equal  in  all 
respects.  Hence  AE  is  equal  to  BE;  also  the  angle 
AEC  is  equal  to  the  angle  BEC ;  therefore  each  is  90°. 

31 


32  '  Geometry. 

In  practice  it  is  not  necessary  to  draw  the  lines  AC, 
BC,  AD,  BD,  CD.  Having  found  the  points  C  and  D, 
placing  the  ruler  on  these  points,  we  may  mark  the  point 
E  in  AB. 

Subsequently,  when  the  subject  of  parallel  lines  comes 
to  be  dealt  with,  another  and  possibly  readier  way  of 
finding  the  middle  point  of  a  line  will  be  given. 

A  number  of  exercises  should  now  be  given  in  bi- 
secting lines  of  different  lengths,  the  dividers  being  used 
in  each  case  to  determine  whether  the  point  reached  is 
the  middle  point. 

It  is  suggested  that  the  pupil  be  given  exercises  in 
estimating  with  the  eye  the  middle  points  of  lines  of 
various  lengths,  these  points  being  afterwards  accurately 
determined  by  geometrical  construction. 

2.  To  bisect  an  angle. 

Let  BAC  be  the  angle.  Place  one  of 
the  points  of  the  dividers  or  compasses 
at  A,  and  mark  off  equal  lengths  AD, 
AE  in  AB  and  AC.  With  centres  D 
and  E  describe  portions  of  circles  with 
equal  radii,  intersecting  at  F.  Then 
drawing  AF,  the  angle  is  bisected  by  it. 

For,  adjusting  the  bevel  to  either  of  the  angles  at  A,  it 
will  be  found  equal  to  the  other. 

Or  again,  we  would  conclude  that  the  angles  at  A  are 
equal  from  the  symmetry  of  the  figure  with  respect  to 
the  line  AF — the  figure  on  one  side  of  this  line  being 
just  the  same  as  the  figure  on  the  other  side,  but  turned 
in  the  opposite  direction. 


Bisection  of  Lines  and  Angles. 


33 


Or  again,  we  may  prove  the  equality  of  the  angles  as 
follows:  The  triangles  DAF,  EAF  have  their  sides 
equal.  Hence  (Ch.  III.,  1)  the  angles  DAF,  EAF  are 
equal. 

In  practice  it  is  not  necessary  to  draw  the  lines  DF, 
EF. 

A  number  of  exercises  should  be  given  in  bisecting 
angles  of  various  magnitudes,  the  bevel  being  used  in 
each  case  to  determine  whether  the  bisection  is  accurate. 

The  protractor  may  also  be  used  for  bisecting  angles. 

It  is  suggested  that  the  pupil  b6  given  exercises  in 
estimating  with  the  eye  the  bisecting  lines  of  a  number 
of.  angles,  the  bisection  being  afterwards  accurately 
reached  by  geometrical  construction. 

Greater  accuracy  is  likely  to  be  secured  in  bisecting 
an  angle,  by  making  AD,  AE  and  DF,  EF  of  consid- 
erable length.  The  point  F  is  then  remote  from  A, 
and  any  trifling  error  in  locating  the  exact  point  where 
the  circles  intersect,  has  less  effect  on  the  angle  at  A 
through  being  on  the  circumference  of  a  large  circle 
(radius  AF). 

3.  From  a  point  in  a  line  to  draw  a  line  at 
right  angles  to  it. 

If  C  be  the  point  in  AB  from 
which -the  perpendicular  is  to  be 
drawn,  place  one  point  of  the  divid- 
ers or  compasses  at  C,  and  mark  off 
equal  lengths  CD  and  CE.  Then 
w^th  centres  D  and  E  describe  por- 
tions of  circles  with  equal  radii,  intersecting  at  F. 
Draw  FC:  it  is  perpendicular  to  AB. 


34 


Geometry. 


For,  applying  the  set-square  or  protractor,  the  angles 
at  C  will  be  found  to  be  right  angles. 

Or  again,  from  the  symmetry  of  the  figure  with 
respect  to  CF,  we  may  conclude  that  the  angles  at  C 
are  right  angles. 

Or  again,  since  the  sides  of  the  triangles  DCF,  ECF 
are  equal,  therefore  (Ch.  III.,  1)  these  triangles  are 
equal  in  all  respects,  and  the  angles  at  C  are  equal. 
Hence  the  angles  at  C  are  right  angles. 

In  practice  the  lines  FD  and  FE  need  not  be  drawn. 

A  number  of  exercises  should  be  given  in  drawing 
lines  at  right  angles  to  others  from  points  in  the  lat- 
ter, the  correctness  of  the  constructions  being  tested 
by  using  the  set-square  or  protractor. 

In  future,  in  the  various  constructions  that  are  to 
be  made,  where  a  line  is  to  be  drawn  at  right  angles 
to  another  from  a  point  in  the  latter,  the  set-square  or 
protractor  should  in  general  be  used  instead  of  this 
construction. 


4.  To  draw  a  line  perpendicular  to   another 
from  a  point  without  the  latter. 

Let  C  be  the  point  without  AB 
from  which  the  perpendicular  is 
to  be  drawn  to  AB.  With  C  as 
centre  describe  a  circle  cutting 
AB  in  D  and  E.  With  D  and  E 
as  centres  describe  portions  of 
circles  with  equal  radii,  intersect- 
ing at  F.  Join  CF,  cutting  AB 
in  G.     CG  is  the  perpendicular  from  C  on  AB 


Bisection  of  Lines  and  Angles.  35. 

For,  applying  the  set-square  or  protractor,  the  angles 
at  G  will  be  found  to  be  right  angles. 

Or  again,  from  the  symmetry  of  the  figure  with 
respect  to  CF,  we  may  conclude  that  the  angles  at  G 
are  right  angles. 

Or  again,  since  the  sides  of  the  triangles  DCF, 
ECF  are  equal,  therefore,  (Ch.  III.,  1)  the  angles  at 
C  are  equal.  Also  since  in  the  triangles  DCG,  EGG 
the  angles  at  C  are  equal,  and  the  sides  about  these 
angles  equal,  therefore  (Ch.  III.,  2)  these  triangles  are 
equal  in  all  respects,  and  the  angles  at  G  are  equal. 
Hence  the  angles  at  G  are  right  angles. 

In  practice  the  lines  CD,  CE,  FD,  FE,  GF  need  not 
be  drawn. 

A  number  of  exercises  should  be  given  in  drawing 
lines  perpendicular  to  others  from  points  without  the 
latter,  the  correctness  of  the  constructions  being  tested 
by  using  the  set-square  or  protractor. 

5.  In  future,  where  a  line  is  to  be  drawn  perpen- 
dicular to  another  from  a  point  without  the  latter, 
the  set-square  or  protractor  should  in  general  be  used 
instead  of  the  preceding  construction.  When  for  this 
purpose  the  protractor  is  used,  the  edge  of  the  ruler 
is  to  be  placed  over  the  centre-point  of  the  protractor 
and  over  the  90°  markj  the  base  of  the  protractor  is 
then  to  be  slid  along  the  line  until  the  edge  of  the 
ruler  is  over  the  given  point  without  the  line.  The 
centre-point  of  the  protractor  then  marks  the  foot  of 
the  perpen<Jicular  on  the  line. 


36 


Geometry. 


The  annexed  diagram  illustrates 
how,  by  sliding  the  set-square  along 
the  ruler,  lines  may  be  drawn  parallel 
to  each  other  5  and  also  how  a  line 
may  be  drawn  perpendicular  to  an- 
other from  any  point,  whether  the 
point  be  without  or  on  the  latter 
line.  In  drawing  a  perpendicular 
to  a  line  by  placing  an  edge  of  the 
right  angle  of  set-square  against  the  latter,  we  are 
often  unable  to  bring  the  perpendicular  up  to  the  line 
by  reason  of  the  right  angle  of  the  set-square  having 
become  rounded  through  use. 

Sometimes  a  convenient  way  of  drawing  a  perpen- 
dicular through  a  point  is  to  draw  a  perpendicular  in 
any  position,  and  then  a  parallel  to  this  through  the 
given  point,  the  former  perpendicular  being  afterwards 
erased. 

Draw  a  number  of  equal  and 
perpendicular  lines  AB,  BC,  CD, 

MN,    and   finally   draw 

AO  perpendicular  to  AB,  and 
NO  perpendicular  to  MN,  as 
indicated  in  the  figure.  The 
accuracy  of  the  series  of  con-  b 
structions  may  be  tested  by 
the  conditions  that  AO  is  both  ^ 
equal  and  perpendicular  to  NO. 


M       I 

^ 

F 

D      E 

Exercises.  37 

Exercises. 

1.  What  is  meant  by  the  distance  of  a  point  from  a  line  ? 

2.  AOB  is  any  angle  and  OC  bisects  it.  What  do  you  observe  as 
to  the  distances  of  any  point  in  OC  from  OA  and  OB  ?     Give  proof. 

What  do  you  observe  as   to  the  angles  which   OC  makes  with 
these  distances  to  OA  and  OB  ?     Give  proof. 

3.  AB  and  CD  intersect  in  O.  Bisect  the  angles  AOC  and  BOD 
by  OE  and  OF.  What  position  do  OE  and  OF  occupy  with  respect 
to  each  other  ?     Give  proof. 

Bisect  the  angle  AOD  by  OG.     What  position  do  OG  and  OE 
or  OF  occupy  with  respect  to  each  other  ?     Give  proof. 

4.  Construct  a  triangle  ABC,  and  find  a  point  in  the  base  BC  such 
that  the  perpendiculars  from  it  on  AB  and  AC  are  equal. 

5.  Taking  any  two  points,  A  and  B,  in  the  plane  of  the  pap^, 
draw  a  line  such  that  the  distances  from  any  point  on  it  k)  A  and  B 
are  equal. 

6.  Find  a  point  equidistant  from  two  given  points  A  and  B,  and 
one  inch  from  a  third  given  point  C.  Is  it  always  possible  to  do 
this? 

7.  In  a  given  straight  line  find  a  point  which  is  equidistant  from 
two  given  points  not  lying  in  the  line. 

8.  Take  two  points,  A  and  B,  in  the  plane  of  the  paper,  and 
describe  a  circle  of  radius  two  inches  which  shall  pass  through  A 
and  B. 

9.  Construct  a  triangle  ABC  with  sides  4,  3|  and  3  inches.  Bisect 
the  angles  ABC,  ACB  by  BD,  CD  meeting  in  D.  What  do  you 
observe  as  to  the  distances  of  D  from  the  three  sides  ?     Give  proof. 

10.  Construct  a  triangle  ABC.  Bisect  AB  in  D,  and  AC  in  E. 
Join  DE,  producing  it  both  ways.     In  the  base  BC,  or  BC  produced, 

take  points  F,  G,  H,  .     .     .     ,  and  join  AF,  AG,  AH 

What  do  you  observe  as  to  the  division  of  the  lines  AF,  AG,  AH 
.     .     .     by  DE  or  DE  produced  ? 

11.  Draw  two  straight  lines  AB,  AC,  and  with  the  set-square 
draw  any  two  lines  at  right  angles  to  them.  What  relation  do  you 
observe  between  the  acute  angle  between  the  lines  and  the  acute 
angle  between  the  perpendiculars  ?    Give  proof. 


38  Geometry. 

12.  Construct  a  triangle  one  of  whose  angles  is  a  right  angle. 
How  many  degrees  do  you  find  in  the  sum  of  the  other  two  angles  ? 
Give  reason. 

13.  Construct  a  triangle  ABC  with  C  a  right  angle.  At  C  make 
the  angle  BCD  equal  to  the  angle  CBA,  CD  meeting  AB  in  D.  What 
do  you  note  as  to  the  magnitudes  of  the  lines  DA,  DB,  DC  ?  Give 
reason. 

14.  ABC  is  an  isosceles  triangle  with  AB  =  AC.  Produce  BA  to 
D,  making  AD  equal  to  AB  or  AC.  Join  CD.  What  is  the  magni- 
tude of  the  angle  BCD  ?     Give  reason. 

15.  AB  and  CD  are  any  two  straight  lines.  Find  a  point  E  such 
that  EAB  and  ECD  are  both  isosceles  triangles. 

16.  With  ruler  and  compasses  {i.e.,  not  using  set-square)  draw 
from  a  point  at  the  extremity  of  a  given  line  another  line  at  right 
angles  to  it,  without  producing  the  given  line. 

17.  Construct  an  equilateral  triangle  ABC,  with  side  two  inches. 
Draw  AD  to  the  bisection  of  the  base  BC.  How  many  degrees  are 
there  in  each  of  the  angles  of  the  triangle  ABD  ? 

18.  Construct  an  equilateral  triangle  ABC,  and  draw  AD  perpen- 
dicular to  the  base  BC.  On  AD  describe  another  equilateral  triangle 
EAD.  How  many  degrees  does  each  of  the  sides  of  EAD  make  with 
each  of  the  sides  of  ABC  ? 

19.  At  the  points  A  and  B  in  the  line  AB  draw  equal  lines  AC,  BD 
at  right  angles  to  AB  and  on  the  same  side  of  it.  Join  CD,  and  pro- 
duce it  and  AB  both  ways.  From  other  points  E,  F,  G  .  .  .  in 
AB  draw  perpendiculars  EK,  FL,  GM  ....  to  CD.  Com- 
pare the  lengths  of  EK,  FL,  GM  .  .  .  with  AC  or  BD.  What 
are  the  angles  at  E,  F,  G     .     .     .     ? 

20.  Construct  a  triangle  ABC,  and  bisect  the  sides  BC,  CA,  AB  at 
D,  E  and  F  respectively.  Join  DE,  EF,  FD.  What  relations  exist 
between  the  lengths  of  the  sides  of  the  triangles  ABC  and  DEF? 
What  relations  exist  between  the  angles  of  the  two  triangles  ?  Make 
three  different  figures,  the  triangles  being  of  different  shapes,  and 
examine  whether  the  same  relations  exist  in  the  three  cases. 


CHAPTER  V. 

Respecting:  Angrles  of  a  Triangrle. 

1.  We  have  seen  (Ch.  III.,  4)  that  the  sum  of  the 
three  angles  of  any  triangle  is  two  right  angles,  or  180°. 

Definition:  In  any  rectilineal  figure,  an  exterior 
angle  is  an  angle  contained  by  any  side  and  an  ad- 
jacent side  produced. 

Produce  the  side  BC  to  D. 
With  the  bevel  or  protractor 
lay  off  the  angle  ABE  equal  to 
the  angle  at  A.  Using  the 
bevel,  examine  now  the  re- 
lation existing  between  the  angle  EBC,  which  is  equal 
to  the  sum  of  the  angles  A  and  B  of  the  triangle, 
and  the  exterior  angle  A  CD. 

Repeat  the  construction  and  examination  in  the  case 
of  a  triangle  of  different  shape,  say  one  in  which  the 
angle  at  B  is  an  obtuse  angle. 

If  the  constructions  and  measurements  have  been 
accurately  made,  it  will  be  found  that  the  exterior 
angle  (ACD)  is  equal  to  the  sum  of  the  two 
interior  and  opposite  angles  at  A  and  B. 

We  may  show  that  this  is  always  the  case  as  follows : 
The  three  angles  of  the  triangle  make  up  180°  j  but 
the  angles  ACB,  ACD  also  make  up  180';  hence 
the  angle  ACD  must  be  equal  to  the  sum  of  the  angles 
A  and  B. 

39 


40 


GrEOMETEY. 


2.  Lay  off  about  a  point,  and 
adjacent  to  one  another,  angles 
equal  to  the  three  exterior  angles 
of  the  triangle;  or,  with  the  pro- 
tractor, measure  the  number  of  de- 
grees in  each  of  these  angles. 
What  is  their  sum  ?  Give  reasons 
for  this  sum  being  what  it  is. 

3.  Of  course,  since  the  exterior  angle  of  any  triangle 
is  equal  to  the  sum  of  the  two  interior  and  opposite 
angles,  it  follows  that  the  exterior  angle  of  any 
triangle  is  greater  than  either  of  the  interior 
and  opposite  angles. 

Also,  since  the  sum  of  the  three  angles  of  any  tri- 
angle is  equal  to  two  right  angles,  the  sum  of  any 
two  angles  of  a  triangle  is  less  than  two  right 
angles. 

We  may,  however,  by  elongating  the  triangle,  make 
the  sum  of  two  of  its  angles  but  little  short  of  two 
right  angles.      Thus   the    Ap^...^^ 

sum  of  A  and  B   will  be      \  " >«_,._^_^ 

found,  on  using  the  pro-     B'  ^ 

tractor,  to  be  but  little  less  than  180°;  and,  by  still 
further  removing  C,  we  may  still  further  increase  their 
sum. 

4.  Construct  a  triangle  with  sides  50,  70  and  90 
millimeters;  and,  by  adjusting  the  bevel  to  the  angles, 
find  out  which  is  the  greatest  angle,  which  is  next  in 
magnitude,  and  which  is  least. 

Repeat  the  same  examination  in  the  case  of  the  tri- 
angle whose  sides  are  2,  4  and  5  inches. 


Respecting  Angles  of  a  Triangle.        41 

What  position  do  you  observe  the  greatest,  inter- 
mediate and  least  angles  occupy  with  respect  to  the 
greatest,  intermediate  and  least  sides  respectively? 

"We  shall  find  for   all  triangles  a  definite   answer  to 
the   preceding   question   in  the   following  proof:     Let 
AC  be  greater  than  AB,  and  let  AD        ^^ 
be  equal  to  AB.     Then  (Ch.  II.,  2) 
the  angles  ABD  and  ADB  are  equal. 
But  the  angle  ABC  is  greater  than   ^  ^ 

ABD,  and  the  angle  ACB  is  less  than  ADB  (Ch.  V.,  3). 
Therefore  the  angle  ABC  is  greater  than  the  angle  ACB. 
That  is,  in  any  triangle,  the  greater  side  has 
the  greater  angle  opposite  it. 

5.  Construct  a  triangle  with  angles  40',  60°,  80°,  and, 
using  the  dividers  or  compasses,  arrange  the  sides  in 
order  of  magnitude. 

Make  the  same  examination  in  the  case  of  a  triangle 
whose  angles  are  100°,  50°,  30°. 

What  position  do  you  observe  the  greatest,  inter- 
mediate and  least  sides  occupy  with  respect  to  the 
greatest,  intermediate  and  least  angles  respectively? 

We  shall  find  for  all  triangles  a  definite  answer  to 
the  preceding   question   in  the   following  proof:      Let 
the  angle  ABC  be  greater  than  the  angle        . 
ACB;    then  the   side   AC  is  greater  than 
the  side  AB.     For,  with  bevel  or  protrac- 
tor,   construct  the   angle  CBD   equal  to 
the  angle  ACB,  so  that  DBC  is  an  isosceles    "  ^ 

triangle.  Then  AC  =  AD  -f  DC  =  AD  +  DB  >  AB,  the 
straight  line  AB  being  the  shortest  distance  between 
A  and  B.  Hence  in  any  triangle  the  greater 
angle  has  the  greater  side  opposite  to  it. 


42  Geometry. 


Exercises. 

1.  Construct  a  quadrilateral  figure.  With  the  protractor  measure 
the  number  of  degrees  in  each  of  the  angles,  and  add  them.  What 
is  the  sum  ?  Deduce  this  sum  also  from  geometrical  truths  already 
reached. 

2.  Produce  the  sides  of  thq  quadrilateral,  and  measure  the  exterior 
angles.  What  is  their  sum?  Deduce  this  also  from  knowing  the 
sum  of  the  interior  angles, 

3.  Construct  a  polygon  with  any  number  of  sides,  ABCDE  .... 
Taking  the  sides  in  order,  produce  each  from  the  preceding  angle,  as 
in  the  figure  of  2,  Ch.  V.  Placing  your  pencil  along  AB,  turn  it 
through  the  exterior  angle  at  B  into  coincidence  with  BC ;  then 
through  the  exterior  angle  at  C  ;  and  so  on,  until  it  has  been  turned 
through  all  the  exterior  angles. 

How  much  has  the  pencil  been  turned  ?     What,  therefore,  do  you 
conclude  the  sum  of  all  the  exterior  angles  of  any -polygon  is? 
Verify  this  by  measureni,ent  with  protractor. 

4.  From  the  result  reached  in  the  previous  question,  show  that  all 
the  interior  angles  of  any  polygon  are  equal  to  twice  as  many  right 
angles  as  the  figure  has  angles  (or  sides),  less  four  right  angles. 

5.  How  many  right  angles  is  the  sum  of  all  the  angles  in  a  pentagon 
(5  sides)  equal  to  ?  If  the  angles  be  equal,  how  many  decrees  are 
there  in  each  ?  , 

6.  How  many  right  angles  is  the  sum  of  all  the  angles  in  a  hexagon 
(6  sides)  equal  to  ?  If  the  angles  be  equal,  how  many  degrees  are 
there  in  each  ?  . 

7.  Construct  an  isosceles  triangle  ABC  (ABp^AC).  In  AB  take 
any  point  D.  With  the  dividers  or  compasses  determine  whether  D 
is  nearer  to  B  or  to  C.     Give  reason. 

8.  ABCD  is  a  right-angled  equilateral  four-sided  figure.  AC  is 
joined.  Any  point  E  is  taken  within  the  triangle  ABC.  Is  E  nearer 
to  B  or  to  D  ?     Give  reasons. 

9.  A  triangle  can  have  only  one  angle  either  equal  to  or  greater 
than  a  right  angle,  i.e.,  ait  least  two  of  the  angles  of  a  triangle  must 
always  be  acute  angles. 

10.  The  perpendicular  is  the  least  line  that  can  be  drawn  from  a 


EXEKCISES.  43 

given  point  to  a  given  line  ;  and  any  line  nearer  to  the  perpendicular 
is  less  than  one  more  remote. 

11.  ABCD  is  a  four-sided  figure.  How  does  the  sum  of  the 
exterior  angles  at  A  and  C  compare  in  magnitude  with  either  of  the 
injberior  angles  B  or  D  ?     Give  reasons. 

12.  ABC  is  a  triangle,  and  0  is  a  point  within  it.  Is  the  angle 
BOC  greater  than,  equal  to,  or  less  than  the  angle  BAG  ?  Give 
reasons. 

13.  Can  more  than  two  equal  straight  lines  be  drawn  to  a  straight 
line  from  a  point  without  it  ?     Give  reasons. 

14.  Use  tHe  result  obtained  in  the  previous  question  to  show  that  a 
circle  cannot  cut  a  straight  line  in  more  than  two  points. 

15.  In  a  right-angled  triangle  the  hypotenuse  is  the  greatest  side. 

16.  In  the  triangle  ABC  can  you  find  a  point  D,  such  that  AD  is 
equal  to  or  greater  than  the  greater  of  the  sides  AB,  AC  ? 

17.  In  any  triangle  can  you  find  a  point  such  that  the  distance 
from  it  to  any  oAe  of  the  angles  is  equal  to  or  greater  than  the 
greatest  of  the  sides  ? 

18.  Describe  two  circles  with  the  same  centre,  i.e.,  concentric. 
Take,  a  point  A  on  the  circumference  of  one,  and  a-  point  B  on  the 
circumference  of  the  other.  When  will  the  line  AB  be  least  ?  Give 
reasons. 

19.  A,  B,  C  are  three  points  on  a  line,  at  any  intervals  apart. 
Rotate  the  line  about  A  in  a  direction  contrary  to  the  motion  of  the 
hands  of  a  clock  through  30°  ;  i.  e. ,  draw  a  new  line  through  A,  making 
an  angle  of  30°  with  the  original  line,  and  locate  B  and  C  on  it  at 
same  intervals  as  before.  Rotate  the  line  about  B  from  its  new 
position,  in  the  same  direction,  through  20°.  Rotate  the  line  about  C 
from  its  new  position,  in  the  same  direction,  through  15°.  What 
angle  does  the  line  in  its  final  position  make  with  its  original  posi- 
tion ? 

20.  The  same  problem  as  the  preceding,  there  being,  however,  four 
angles,  45°,  60°,  30°  and  90°. 

The  point  in  the  last  two  questions  is  that  if  a  line  rotates 
through  various  angles  and  about  different  points  in  it,  the  aggre- 
gate rotation  is  the  same  as  if  it  all  took  place  about  a  single  fixed 
point  in  the  line. 


CHAPTER  VI, 

Parallel  Lines. 

1.  Parallel  lines  were  defined  to 
be  such  as  have  the  same  direc- 
tion. Thus  the  lines  in  the 
figure,  though  differing  in  position, 
have  all  the  same  direction,  and 
are  parallel. 

2.  AC    and   DE   are    straight       a^ 
lines.      Using   the   bevel,   what 
do  you  observe   with   reference 
to  the  magnitudes  of  the  verti-     ^ 
cally  opposite   angles  ABD   and  CBE?    What  with 
reference  to  the  magnitudes  of  the  angles  ABE,  DBC  ? 

Draw  other  intersecting  straight  lines  and  note  the 
magnitudes  of  vertically  opposite  angles. 

We  may  demonstrate  the  relation  between  such  angles 
as  follows : 

Z ABD+  z^ ABE  =  2  rt.  angles  =  /_'EBC+  ZEBA,  and 
dropping  from  both  sides  the  angle  ABE,  we  have 
LABD=LEBC. 

Hence  if  two  straight  lines  cut  one  another, 
the  vertically  opposite  angles  are  equal. 

Yet  such  a  proposition  scarcely  needs  demonstration  j 
for,  as  was  said  in  Chapter  I.,  a  straight  line  has  the 
same  direction  throughout  its  entire  length.  Hence 
the  two  lines  ABC,  DBE  must  deviate  from  one  another 
as  much  to  the  left  of  B  as  to  the  right  of  B,  and  thus 
the  angles  ABD,  EBC  are  equal. 

44 


Parallel  Lines. 


45 


3.  Straight  lines  which  deviate  yE. 

from  the    same   straight   line  by  a/_ 

the     same     amount,    i.e.,    which         / 

make     eqnal     angles    with    this     ^ D 

straight   line   in  the   same  direc-   ^ 

tion,  must  have  the  same  direction,  and  therefore  must 

be  parallel. 

Thus  if  the  directions,  or  lines,  AB  and  CD  deviate 
equally  from  the  same  direction,  or  line,  EF,  i.e.,  if  the 
angles  EAB  and  ACD  are  equal,  then  AB  and  CD  have 
the  same  direction,  and  are  said  to  be  parallel. 

ACD  is  said  to  be  the  interior  and  opposite 
angle  with  respect  to  the  exterior  angle  EAB. 

It  is  to  be  noted  that  the  parallel 
lines  are  inclined  to  the  cutting 
line  equally  and  in  the  same  direc- 
tion. Thus  though  AB  and  CD 
deviate  equally  from  EF,  they  deviate  in  opposite  direc- 
tions, and  therefore  are  not  parallel. 

4.  It  is  understood,  then,  that 
if  AB  and  CD  are  any  two  par- 
allel lines,  and  any  line  GH  cuts 
them,  the  exterior  angle  GEB 
is  equal  to  the  interior  and 
opposite  angle  EFD. 

5.  The  angles  AEF,  EFD  are  called  alternate  angles. 
By  actual   measurement,   with  the   bevel  or  protrac- 
tor, show  that  they  are  equal. 

We  may  also  prove  their  equality  thus : 

I  GEB  =  Z.  EFD,  because   the   lines    are  parallel ;  also 

Z. GEB— Z AEF,    because   these  are    vertically    opposite 

angles  J  .-.   ZAEF=ZEFD. 


A    C 


46  GrEOMETRY. 

6.  The  angles  BEF,  EFD  are  called  interior  angles. 
By  measurement   with   the    protractor,  or  by  laying 

off,  with  the  bevel,  two  adjacent  angles  equal  to  them, 
show  that  the  sum  of  BEF  and  EFD  is  180°. 
We  may  also  prove  this  thus : 

ZGEB=ZEFD; 
.-.       z:GEB+  ZBEF=ZEFD-f-  LBEF, 
But   Z GEB-t- BEF  =  2  rt." angles; 
Z  BEF  +  L  EFD  =  2  rt.  angles. 

7.  There    is    no    difficulty    in     verifying    by    actual 
measurement,  or  in  proving  the  following  equalities : 

ZBEF=ZEFC 

LBFI>=  LFEB 

ZAEF+  Z.EFC  =  2  rt.  angles.  ' 

8.  To  draw  a  straight  line  through  a  given  point  A 
parallel  to  a  given  straight  line  BC. 

Through  A  draw  DAE,  cutting  q 

BC,    and   make    the    angle    DAF    q. ^ 

equal  to  the    angle   AEC.      Then 
AF  is   parallel  to   BC.      FA   may    __ 
then    be  produced,    if    necessary,   ^ 
to  G. 

Of  course  we  could  have  drawn  GA  parallel  to  BC  by 
making  the  angle  GAE  equal  to  the  alternate  angle  AEC. 

The  line  through  A  parallel  to  BC  can  also  be  drawn 
without  measuring  any  angle,  as  follows: 

With  A   as    centre    and    radius, 
say,   of   2   inches,    describe  a  por-  ^' 
tion  of   a  circle   cutting  BC  in  D. 

Measure    off    on    this    a    distance         

AE  of   1    inch,    so    that    E  is    the  ^  ^ 


Parallel  Lines.  47 

middle  point  of  AD.  With  centre  E  and  any  radius 
of  sufficient  length  to  reach  BC,  describe  a  portion  of 
a  circle  cutting  BC  in  F ;  and  let  the  diameter  of  this 
circle,  through  F,  meet  the  circle  again  in  G.  Then  AG 
is  parallel  to  BC.  For  the  sides  AE,  EG  of  the  triangle 
AEG  are  equal  to  the  sides  DE,  EF  of  the  triangle 
DEF.  Also  the  angles  AEG,  DEF  are  equal.  Hence 
these  triangles  are  equal  in  all  respects,  and  the  angle 
GAE  is  equal  to  the  angle  EDF.  AG  is  therefore 
parallel  to  BC. 

A  number  of  exercises  should  be  given  pupils  in 
drawing  lines  through  given  points  parallel  to  lines  in 
given  positions,  using  both  the  preceding  methods. 
At  the  end  of  each  construction  the  accuracy  of  the 
work  may  be  tested  with  the  parallel  rulers,  or  with 
ruler  and  set-square  (Ch.  IV.,  5),  or  by  examining 
whether  lines  drawn  perpendicular  to  each  pair  of 
parallels  are  equal  in  length.     (See  9  and  10,  following.) 

For  the  most  part,  in  future,  in  drawing  parallel  lines 
parallel  rulers  are  to  be  used,  or  ruler  and  set-square 
(Ch.  IV.,  5). 

9.  A  straight  line  which  is  perpendicular  to  one  of 
two  parallel  lines,  is  also  perpendicular  to  the  other.   The 

truth  of  this   should  be   tested   by    ^ ^ b 

drawing  with  the  set-square  a  line 
perpendicular  to  one  of  the  parallels,     


and  examining,  with  the  set-square,    ^ 
whether  it  is  also  perpendicular  to  the  other. 

Of  course  this  is  only  a  particular  case  of  the  truth, 
that  parallel  lines  have  each  the  same  direction  with 
respect  to  any  third  line  that  cuts  them. 

Or  we  may  prove  it  as   follows:     If  DFE  is  a  right 


48  GrEOMETRY. 

angle,  then  since  DFE  +  FEB  =  2   rt.  angles,   FEB  mnst 
also  be  a  right  angle. 

10.  Two    parallel  lines  are,  of     /k_e. g         b 

course,  throughout  their  lengths 

at   the  same   distance  from    one 

another.  For,  with  the  set-square  ^  )r\  u 

or  protractor,  draw   lines   EF,  GH,  .  .  .  perpendicular 

to  AB  and  CD.     Then,   if   the   dividers  be  adjusted  to 

the  length  EF,  they   will  be   found   to  be   adjusted  to 

the  other  lengths  GH,  .  .  . 

We  may  prove  that  this  is  always  the  case,  as  fol- 
lows :  EF  and  GH  are  parallel  to  one  another  because 
they  have  the  same  direction  with  respect  to  the  third 
line  AB  (or  CD). 

Again,   L  EGF  =  /_  GFH,  being  alternate  angles ; 
ZEFG=Z.HGF,       "  "  " 

Side  FG  is  common  to  the  two  triangles; 
.-.  (Ch.  Ill,  5)  EF  =  GH. 
"We  have  everywhere  illustrations  of  this.     Thus  we 
say  that  an  ordinary   board  or   ruler,  whose    sides  are 
parallel,  is  of  the  same  width  throughout  its  length. 

11.  The  method  of  drawing  a  line  parallel  to  another 
by  sliding  the  set-square  along  the  ruler  (Ch.  IV.,  5) 
receives  its  justification  in  the  first  paragraph  of  §  3 
of  this  chapter.  The  line  EACF  corresponds  to  the 
edge  of  the  ruler ;  the  lines  AB,  CD  to  the  edge  of  the 
set-square  in  its  two  positions;  and  the  angles  EAB,  ECD 
to  the  angle  of  the  set-square  in  its  two  positions,  the 
angle  of  the  set-square  being  of  course  always  the  same. 

It  may  be  added  that,  in  drawing  parallel  lines,  some 
prefer  the  ruler  and  set-square  to  parallel  rulers.  The 
cost  of  an  instrument  is  saved.     If  the  edges  of  ruler 


Exercises.  49 

and  set-square  are  perfectly  straight,  the  method  gives 
absolutely  correct  results.  Parallel  rulers  possibly  work 
more  rapidly  and  conveniently. 

Exercises. 

1.  Draw  a  line  through  A  parallel  to  a  line  BC,  as  follows  :  Join 
AB,  AC.  With  B  as  centre,  and  radius  equal  to  AC,  describe  a  circle. 
With  C  as  centre,  and  radius  equal  to  AB,  describe  a  circle.  Let  D 
be  the  point  where  the  circles  intersect  on  the  same  side  of  BC  as  A. 
Then  AD  is  parallel  to  BC. 

Test  with  parallel  rulers. 

Examine  the  equality  of  alternate  angles. 

Examine  whether  the  sum  of  interior  angles  on  the  same  side  is 
180°. 

Prove  that  the  alternate  angles  ADB,  DBC  are  equal,  and  that, 
therefore,  the  lines  are  parallel. 

2.  If  AB,  CD  intersect  in  O,  and  AO  =  OB,  andCO  =  OD,  what 
position  do  AD,  CB  occupy  with  respect  to  each  other;  and  what  do 
AC  and  DB  ?    Apply  tests  with  instruments.    Give  reasons,  i.e., proof. 

3.  If  AB,  CD  intersect  in  O,  and  AO  =  OD,  CO=:OB,  what  position 
do  AD,  CB  occupy  with  respect  to  each  other  ?  Apply  tests.  Give 
reasons. 

4.  Construct  a  quadrilateral  with  two  sides  equal,  and  the  other 
two  parallel  and  unequal. 

5.  In  the  receding  question  produce  the  equal  sides  to  meet,  and  by 
applying  tests  determine  the  character  of  the  two  triangles  so  formed. 

6.  The  two  interior  angles  on  the  same  side  which  one  line  makes 
with  two  others  are  105°  and  70°.  Infer  from  6  of  Ch.  VI.  that  the 
lines  meet.     On  which  side  of  the  cutting  line,  and  why  ? 

7.  AD  and  BF  are  parallel  lines.  From  A  draw  equal  lines  AB, 
AC  to  BF;  and  also  equal  lines  DE,  DF,  less  than  the  former.  Show 
that  AC  meets  DE  and  DF  on  one  side  of  the  parallel  lines,  and  AB 
meets  them  on  the  other  side. 

8.  A,  B  are  the  extremities  of  the  diameter  of  a  circle,  and  par- 
allel lines  AC,  BD  are  drawn,  terminated  by  the  circle.  What  is  the 
relation  of  AC,  BD  as  to  magnitude  ?     Give  reasons. 

9.  A  is  a  point  not  lying  in  the  straight  line  BC.  From  A  draw 
lines  AD,  AE,  AF  ...  to  BC,  and  produce  them  to  K,  L,  M.  - 


50  Geometry. 

making  DK  =  AD,  EL  =  AE,  FM  =  AF,  .  .  .  .     What  do  you  observe 
as  to  the  positions  of  the  points  K,  L,  M,  .  .  ?     Give  reasons. 

10.  Two  parallel  lines  are  3  inches  apart,  and  a  point  A  is  taken  2 
inches  from  one  line  and  1  inch  from  the  other.  Lines  are  drawn 
through  A  terminated  by  the  parallels.  By  measurement  determine 
how  these  lines  are  divided  at  A. 

11.  Construct  a  triangle  with  sides  2,  3  and  4  inches.  Bisect  the 
sides  and  join  the  points  of  bisection.  What  do  you  observe  as  to 
the  direction  of  the  sides  of  the  new  triangle  ?  What  as  to  magni- 
tudes of  its  angles  and  sides  ? 

12.  Construct  a  triangle,  and  through  its  angular  points,  with  the 
parallel  rulers  draw  lines  parallel  to  the  opposite  sides.  Four  new 
triangles  are  thus  constructed.  Compare  their  sides  and  angles  with 
those  of  the  original  triangle,  and  give  results  of  comparison. 

13.  Construct  a  triangle  ABC,  and  through  any  points  D,  E,  F  in 
the  plane  of  the  paper  draw  lines  parallel  to  BC,  CA,  AB.  Compare 
the  angles  of  the  new  triangle  with  those  of  the  original. 

14.  If  through  a  point  A  any  two  lines  be  drawn,  and  through  any 
point  B  lines  be  drawn  parallel  to  the  former  two,  prove  that  the 
angles  at  A  and  B  are  equal. 

15.  ABC,  CDE  are  two  triangles  with  AB,  CD  equal  and  parallel, 
and  also  BC,  DE  equal  and  parallel.  What  position  do  AC,  CE 
occupy  with  respect  to  each  other  ? 

16.  Make  an  irregular  drawing  on  the  paper  to  represent  a  pond, 
or  other  obstruction,  and  on  opposite  sides  of  it  take  points  A  and 
B.  By  a  line  construction  about  the  pond,  with  measurements, 
obtain  a  line  at  A  which  if  produced  would  pass  through  B,  without 
placing  the  ruler  on  AB. 

17.  Draw  two  lines,  both  parallel  to  the  same  straight  line.  What 
is  their  position  with  respect  to  each  other  ? 

18.  The  side  BC  of  a  triangle  ABC  is  produced  to  D.  Bisect  the 
angles  BAC,  ACD.  Can  the  bisecting  lines  be  parallel  to  one  another  ? 

19.  On  any  line  AC  as  diagonal,  construct  a  quadrilateral  ABCD 
with  its  opposite  sides  equal.  How  are  the  opposite  sides  placed  with 
respect  to  each  other  ?     Test  and  give  proof. 

20.  Two  lines  make  an  angle  of  63°  with  each  other.  Place  a 
straight  line  2  inches  long  with  its  ends  resting  on  them,  and  making 
an  angle  of  80°  with  one  of  them. 


CHAPTER  VII. 

Parallelogrrams,  Rectangles  and  Squares. 

1.  With  the  parallel  rulers,  or 
by  other  means,  draw  a  pair  of 
parallel  lines  AB,  CD,  and  also 
another  pair  of  parallel  lines  EF, 
GH,  inclined  to  the  former  pair 
at  any  angle. 

The   figure    KLMN    is   called   a 
parallelogram,   i.e.,   a  parallelogram  is   a  four- 
sided  figure  whose  opposite  sides  are  parallel. 

With  the  dividers  compare  the  lengths  of  KL  and 
NM;  also  the  lengths  of  KN  and  LM.  With  the  bevel 
compare  the  magnitudes  of  the  angles  NKL  and  NML ; 
also  the  magnitudes  of  the  angles  KLM  and  KNM. 

Construct  two  or  more  parallelograms  with  sides  of 
different  lengths,  and  angles  of  different  magnitudes; 
and  in  the  case  of  each  compare  the  magnitudes  of 
the  opposite  sides  and  angles. 

The  result  of  such  observations  will  be  that  the 
Opposite  sides  and  angles  of  parallelograms  are 
equal. 

We  may  also  prove  this  as  follows:     Draw  KM,  the 
diameter  or  diagonal,  as  it  is  called,  of  the  parallelo- 
gram.    We  have  in  the   two   triangles  NKM,  LMK, — 
The  side  KM  common  to  both, 
the  alternate   angles  NKM,  LMK  equal, 
"  ''  ^'        NMK,  LKM      " 

51 


52  Geometey. 

Hence  (CL  III.,  5)  these  triangles  are  equal  in  all 
respects,  i.e., 

KN  =  ML, 
KL  =  MN, 
ZKNM=ZKLM. 
Also   ZNKM=ZLMK, 
and     ZLKM^ZNMK; 
therefore  adding  ^NKL=ZNML. 
Of    course   the    triangle   KNM,    if   cut   out,   can   be 
fitted   on  the   triangle   MLK,    and   is  equal   to  it,  i.e., 
the  diagonal  of  a  parallelogram  bisects  it. 

2.  Draw  a  pair  of  parallel  lines, 
AB    and    CD.       In   AB    take    any   ^ 
length    KL,     and,     adjusting    the    ^ 
dividers  to  it,  in  CD   mark  off  an 
equal  length  MN.     Join  K,  M  and 
L,  N. 

Using  the  dividers,  what  do  you  note  with  reference 
to  the  lengths  of  KM  and  LN  ?  Using  the  bevel  or  par- 
allel rulers,  what  do  you  note  with  reference  to  the 
position  of  KM  and  LN    with  respect  to  one  another? 

Draw  other  parallel  lines,  mark  off  on  them  equal 
lengths,  join  the  extremities  of  these  equal  lengths, 
and  repeat  the  examination  as  to  the  lengths  and  rela- 
tive position  of  the  joining  lines. 

The  result  of  such  observations  will  be  that  the 
straight  lines  joining  the  extremities  of  equal 
and  parallel  straight  lines  are  themselves 
equal  and  parallel. 

We  may  prove  this  as  follows:  In  the  triangles 
LKN,  MNK,  the  sides  LK,  KN  are  equal  to  the  sides 
MN,  NK  ;  and  the  angles  LKN,  MNK  are  equal.     Hence 


E   C 


Pakallelogkams,  Rectangles  and  Squakes.  53 

these  triangles  are  equal  in  all  respects.  Therefore 
LN  and  MK  are  equal.  Also  the  alternate  angles  LNK 
and  MKN  are  equal,  and  therefore  LN  and  MK  are 
parallel. 

3.  With  the  power  of  drawing  parallel  lines  we  have 
another  means  of  bisecting   a  line,  indeed  of  dividing 
a    line    into    any    number  of 
equal  parts : 

Let  AB  be  the  line  to  be  bi- 
sected. Draw  through  A  any  a- 
other  line  AC,  and  with  the 
dividers  mark  off  on  it  equal  lengths  AD,  DE.  Join 
BE,  and  with  the  parallel  rulers  draw  DF  parallel  to 
BE.  F  is  the  bisection  of  AB.  For,  drawing  FG  par- 
allel to  AC,  the  triangles  ADF,  FGB  are  evidently  equal, 
and  AF  is  equal  to  FB. 

In  employing  this  method  of  bisecting  a  line,  we 
may  avoid  altogether  drawing  the  lines  AC,  &c.  For, 
place  the  edge  of  the  ruler  against  A,  and,  close  to  the 
edge  of  the  ruler,  with  the  sharp  points  of  the  dividers, 
mark  the  points  D  and  E  (the  distances  AD,  DE  being 
equal).  Then  place  the  edge  of  the  parallel  rulers 
against  the  points  B  and  E,  and  move  the  edge, 
parallel  to  itself,  back  to  D.  The  point  in  which  the 
edge  cuts  AB  is  its  middle  point,  and  can  be  marked 
with  a  point  of  the  dividers. 

It  is  well  to  so  place  AC  and  the  points  D  and  E, 
that  the  lines  DF,  EB  cut  AB  at  nearly  90°.  The 
point  F  is  thus  located  with  most  deflniteness. 

We  leave  to  the  pupil  to  discover  for  himself,  fol- 
lowing the  suggestion  here  given,  a  means  of  dividing 
a  straight  line  into  any  number  of  equal  parts. 


54 


Geometry. 


4.  Having  drawn  two  parallels, 
adjust  the  points  of  the  divid- 
ers to  a  distance  of,  say,  1 
inch  from  one  another.  Place 
one  point  at  A,  and  let  the  other 

point  of  the  dividers  meet  the  other  parallel  at  B. 
Then  J  inch  from  A  gives  the  middle  point  of  AB. 
Draw  a  number  of  lines  through  C,  and  terminated 
by  the  parallels.  Using  the  dividers,  C  will  be  found 
to  be  the  middle  point  of  all  these  lines. 

5.  If  the  angles  of  a  parallelogram 
are  right  angles,  it  is  called  a  rect- 
angle. 


If  the  adjacent  sides,  and 
therefore  all  the  sides,  of  a 
parallelogram  are  equal,  it 
is  called  a  rhombus. 


If  the  angles  of  the  rhombus  are 
right  angles,  the  figure  is  called  a 
square,  i.e.,  a  square  is  a  four- 
sided  figure  with  all  its  sides  equal, 
and  all  its  angles  right  angles. 

Construct  the  following  parallelo- 
grams : 

Sides  50  and  80  millimetres,  and  included  angle  45°. 

Sides  40  and  110  millimetres,  and  included  angle  110°. 

Sides  2  and  3  inches,  and  included  angle  58°. 

In  all  cases  test  the    equality  of   the   opposite    sides 
and  angles. 


EXEKCISES.  55 


Construct  the  following  rhombuses: 

Sides  70  millimetres,  and  one  angle  60° 

Sides  5  inches,  and  one  angle  75°. 

Sides  3 J  inches,  and  one  angle  15°. 


Construct    the   square    whose  side  is  50  millimetres  j 
whose  side  is  4 J  inches;  whose  side  is  70  millimetres; 


Exercises. 

1.  With  two  equal  triangles,  cut  out  of  paper,  form  a  parallelo- 
gram. 

2.  Draw  a  number  of  straight  lines  of  various  lengths,  and,  by  the 
method  of  §  3,  bisect  them,  using  points  only  in  your  construction. 
With  the  dividers  test  the  accuracy  of  your  construction. 

3.  Draw  a  number  of  straight  lines  of  various  lengths,  and,  by  the 
method  of  §  3,  trisect  them,  using  points  only  in  your  construction. 
With  the  dividers  test  the  accuracy  of  your  construction, 

4.  Draw  both  diagonals  in  a  number  of  parallelograms,  and  examine 
how  the  point  in  which  the  diagonals  intersect  divides  them.  Give 
proof. 

5.  Draw  two  lines  whose  intersection  bisects  both,  and  show  by 
using  parallel  rulers  that  the  lines  joining  the  extremities  of  the 
bisected  lines  are  parallel  in  pairs.     Give  proof. 

6.  Two  equal  and  parallel  lines  are  joined  towards  opposite  parts. 
How  do  the  joining  lines  divide  each  other  ?  Apply  tests.  Give 
proof. 

7.  With  compasses  and  ruler  only,  construct  a  four-sided  figure 
with  opposite  sides  equal.  How  are  opposite  sides  placed  with  respect 
to  each  other  ?     Apply  tests.     Give  proof. 

This  exercise  explains  the  principle  of  the  construction  of  parallel  rulers. 

8.  With  protractor  and  ruler,  construct  a  four-sided  figure  with 
one  pair  of  opposite  angles  equal  and  each  75°,  and  the  other  pair  of 
opposite  angles  equal  and  each  105°,  What  is  the  figure?  Apply 
tests.     Give  proof. 


56  Geometry. 

9.  Can  you  construct  a  four-sided  figure  with  opposite  angles  equal, 
such  pairs  of  angles  having  any  magnitude  ?  If  there  be  any  restric- 
tion, what  is  it  ? 

10.  Show  how  to  bisect  a  straight  line  by  means  of  a  set-square  (or 
other  triangular  shape)  and  ruler. 

11.  Using  protractor  and  ruler,  on  a  given  diagonal  AB  construct 
a  four-sided  figure,  such  that  AB  bisects  the  angles  at  A  and  B,  these 
angles  being  equal.     What  is  the  figure  ?     Apply  tests.     Give  proof. 

12.  Give  proof  that  the  diagonals  of  a  parallelogram  or  rhombus 
are  in  general  unequal.     When  are  they  equal  ? 

13.  At  what  angle  do  the  diagonals  of  a  rhombus  intersect?  Apply 
test.     Give  proof. 

14.  Draw  AB,  CD  intersecting  in  O,  and  make  OA,  OB,  OC,  OD 
all  equal  to  one  another.  What  is  the  figure  OCBD  ?  Apply  tests 
and  give  proof. 

15.  If  two  railway  tracks  of  the  same  gauge  cross  one  another  at 
any  angle,  what  special  kind  of  parallelogram  is  formed  by  the  rails  ? 
Apply  tests.     Give  proof. 

16.  In  the  preceding  question,  if  the  tracks  be  of  different  gauges, 
can  this  special  kind  of  parallelogram  be  formed  ? 

17.  Describe  a  circle,  and  drawing  any  two  diameters,  join  their  ex- 
tremities.   What  is  the  figure  so  formed  ?   Apply  tests  and  give  proof. 

18.  Construct  a  parallelogram  with  angles  120°  and  60°,  and  sides 
110  and  50  millimetres.  Bisect  the  angles  of  the  parallelogram.  What 
is  the  figure  formed  by  the  bisecting  lines  ?    Apply  test.    Give  proof. 

19.  Show  that  every  straight  line  through  the  intersection  of  the 
diagonals  of  a  parallelogram  divides  the  parallelogram  into  two  equal 
areas. 

20.  D  is  any  point  lying  in  the  angle  BAC.  Construct  a  parallelo- 
gram ABEC,  such  that  D  may  be  the  intersection  of  the  diagonals. 

21.  D  is  any  point  lying  in  the  angle  BAC.  Through  D  draw  a 
line  bisected  at  D  and  terminated  by  AB,  AC. 

22.  On  any  line,  with  the  dividers  mark  off  equal  lengths  AB,  BC, 
CD,  DE  .  .  .  .  ;  and  through  A,  B,  C,  D,  .  .  ,  .  draw,  with  the 
parallel  rulers,  in  any  direction,  parallel  lines  cutting  any  line  in 
K,  L,  M,  N,  .  .  .  .  What  do  you  note  as  to  the  lengths  of  KL,  LM, 
MN ? 


CHAPT]^R  VIII. 

Certain  Relations  in  Area  between  Parallelogrrams 
and  Triang-les. 

1.  If  two  parallelograms  have  equal  bases 
and  equal  heights,  or  altitudes,  they  are  equal 
in  area. 

For  if  such  be  placed,  or  constructed,  on  the  same 
base,  we  shall  get  one  of  the  three  following  cases : 

(1)  The  parallelograms  may  lie  a d e. 

as    ABCD    and   DBCE,    and   the 

triangle    EDC    can   be    cut    out, 

pushed  to  the  left,   and  made  to 

cover  exactly  the   triangle  DAB.        b  c 

Thus  the  area  DBCE  is  made  to  coincide  with  the  area 

ABCD,  and  they  are  equal. 

(2)  The  parallelograms    may   lie  as     ^      ^         OF 
ABCD   and   EBCF,    and   the   triangle 
FDC     can    be     cut    out,    pushed    to 
the  left,  and   made  to   cover   exactly 
the    triangle    EAB.      Thus    the    area  B  c 
EBCF  is  made   to   coincide   with  the   area  ABCD,  and 
they  are  equal. 

(3)  The  parallelograms  may  lie  a  l  d  e  m  f 
as  ABCD  and  EBCF.  Draw  GHK 
parallel  to  BC  or  AF.  The  tri- 
angle KHC  can  be  cut  out,  pushed 
to  the  left,  and  made  to  cover 
exactly  the  triangle  HGB. 

Next  draw  GL  parallel  to  BE  or  CF,  and  KM  par- 
allel to  AB  or  CD.  The  figure  EHKM  can  now  be 
cut  out,  pushed  to  the  left,  and  made  to  cover  exactly 

57 


¥ 


E F 


58  Geometey. 

the  figure  LGHD ;  and  the  triangle  FMK  can  be  cut 
out,  moved  to  the  left,  and  made  to  cover  exactly  the 
triangle  LAG.  Thus  the  area  EBCF  is  made  to  coin- 
cide with  the  area  ABCD,  and  they  are  equal. 

If  the   parallelo-     a d 

grams  be  much  in- 
clined from  one 
another,  more  than 
one  line  correspond- 
ing to  GHK  must 
be  drawn.  The  ac- 
companying figure  B  C 
illustrates  how  EBCF  must  be  cut  up  so  that  its  sec- 
tions may  exactly  make  up  ABCD.  Corresponding 
numbers  are  placed  on  the  figures,  which  are  to  be 
placed  on  one  another.  It  will  be  noticed,  however, 
that  all  the  triangles,  on  both  sides,  numbered  from  1 
to  6,  can  be  made  to  coincide  with  one  another. 

Several  pairs  of  parallelograms  should  be  constructed, 
each  pair  with  the  same  base  and  between  the  same 
parallels,  and  the  cutting  just  described  should  be  done 
so  as  to  show  that  each  pair  may  be  made  to  coincide. 
Care  should  be  taken  to  illustrate  the  different  cases 
that  may  occur.  The  figures,  of  course,  must  be  accur- 
ately and  completely  drawn  before  the  cutting  is  pro- 
ceeded with. 

2.  If  any  triangle  be  cut  along  a 
straight  line  through  the  centres  of  two 
of  its  sides,  the  two  parts  of  the  tri- 
angle can  be  formed  into  a  parallelo- 
gram, of  course  equal  in  area  to  the 
triangle.  For,  let  D  and  E  be  the 
middle    points    of    two    sides,    so    that 


Areas  of  Parallelograms  and  Triangles.   59 

the  cutting  is  made  along  DE.  Then  let  the  triangle 
ADE  be  turned  about  E,  through  180°,  in  the  direction 
indicated  by  the  arrow  head.  The  point  A  arrives  at 
C,  and  D  at  F.  Since  the  alternate  angles  ADE,  EEC 
are  the  same,  DB  is  parallel  to  CF,  and  they  are  equal. 
Hence  (Ch.  VII.,  2)  DBCF  is  a  parallelogram. 

Since  D  is  the  middle  point  of  AB,  it  is  (Ch.  VII.,  4) 
mid-way  on  the  perpendicular  through  D  to  each  of 
the  parallels  GAH  and  BC.  Hence  the  triangle  ABC 
has  twice  the  altitude  of  the  parallelogram  DBCF  into 
which  it  has  been  converted. 

3.  If  two  triangles  have  the  same  or  equal 
bases,  and  equal  altitudes,  they  are  equal  in 
area. 

For,  let  the  triangles  ABC, 
GBC,  upon  the  same  base  BC, 
have  the  same  altitude.  The 
triangle  ABC  can,  by  a  sec- 
tion along  DE,  be  converted 
into  the  parallelogram  DBCF, 
whose  altitude  is  half  that  of 
the  triangle.  Also,  the  triangle  GBC  can,  by  a  section 
along  HK,  be  converted  into  the  parallelogram  LBCK, 
whose  altitude  is  haK  that  of  the  triangle.  Hence  the 
parallelograms  DBCF,  LBCK,  being  on  the  same  base 
and  with  equal  altitudes,  are  (Ch.  VIII.,  1)  equal  in 
area.     Hence  the  triangles  are  equal  in  area. 

The  parallelograms  DBCF,  LBCK  may  be  cut  up 
(Ch.  VIII.,  1)  so  that  the  one  exactly  coincides  with 
the  other.  Hence,  in  so  cutting  and  placing  the  par- 
allelograms, the  original  triangles  are  made  to  exactly 
coincide  with  each  other. 


60 


Geometky. 


4.  The  triangle  ABC  is  half  of 
the  parallelogram  ABCD,  and, 
therefore,  half  of  the  rectangle 
HBCG.  Hence  if  we  find  E, 
the  bisection  of  BC,  and  draw 
EF  perpendicular  to  BC,  the  tri- 
angle ABC  is  eqnal  to  either  of 
the  rectangles  HBEF  or  FECG. 

Hence  to  construct  a  rectangle  equal  to  a  tri- 
angle, bisect  the  base  of  the  triangle,  and  on 
the  half-base  construct  a  rectangle  of  the  same 
altitude  as  the  triangle. 

Construct  rectangles  equal  in  area  to  the  following 
triangles : 

Sides  80,  90,  140  miUimetres.  ' 

Sides  70,  100  millimetres,  and  included  angle  50°. 
Base  80  millimetres,  and  angles  at  base  45°  and  75°. 


8 


5.  To  find  the  area  of  a  rectangle 
we  multiply  the  length  by  the 
breadth.  Thus  the  adjoining  rect- 
angle being  8  units  in  length, 
and  5  units  in  breadth,  the  area 
is  evidently  8  x  5  =  40  square  units. 

Since  the  area  of  a  triangle  is  half  that  of  the 
rectangle  on  the  same  base  and  with  same  altitude, 
we  may  find  the  triangle ^s  area  by  multiplying 
the  base  by  the  perpendicular  height  and 
dividing  by  2. 

Calculate  approximately,  in  square  millimetres,  the 
areas  of  the  triangles  in  4,  by  finding  the  lengths  of 
the  sides  of  the  rectangles  which  have  been  constructed 
equal  to  the  triangles. 


Akeas  of  Pakallelograms  and  Triangles.    61 


It  will  be  interesting  for  the  teacher  to  calculate  the 
areas  of  such  triangles  by  the  usual  trigonometrical 
formulse,  that  he  may  inform  the  class  as  to  the  close- 
ness of  their  approximations  reached  by  instrumental 
methods. 

6.  In  the  adjoining  figure 
BED  is  the  diagonal  of  each 
of  the  parallelograms  ABCD, 
FBHE,  KEGD,  and  therefore 
bisects  each  of  them.  Hence 
we  have 

triangle  ABD  =  triangle  CBD, 
"       FBE  =        ''        HBE, 
'^       KED=        ''        GED. 
Therefore  the  parallelogram  AFEK  is  equal  to 
the    parallelogram  CHEG.      Note    the   position   of 
these  parallelograms  with  respect  to  the  diagonal  BD. 

7.  In  constructing  a  rectangle  equal  to  a  given  tri- 
angle (Ch.  VIII.,  4)j  one  of  the  sides  of  the  rectangle 
is  half  the  base  of  the  triangle.  We  may,  however, 
construct  a  rectangle  equal  to  any  triangle,  and  give 
to  one  of  the  sides  of  the  rectangle  any  length  we 
choose. 

Thus  having  constructed 
FECG  equal  to  the  triangle 
ABC,  suppose  we  wish  to  make 
a  rectangle  equal  to  the  tri- 
angle, wdth  -one  of  its  sides  of 
length  CH.  Complete  the  rect- 
angle EKHC,  and  let  KC,  FG 
meet  in  L.  Draw  the  remain- 
ing lines  as  indicated  in  the  figure.  Then  the  rect- 
angle CM,  whose  side  CH  is  of  the  required  length,  is 


AK 

F 

G       y 

B       E 

/ 

C 

62  G-EOMETRY. 

equal  to  the  rectangle  FC  (Ch.  VIII.,  6),  which  is 
equal  to'  the  original  triangle  ABC. 

Construct  rectangles,  each  with  a  side  of  50  milli- 
metres, equal  to  the  triangles  in  4. 

The  sides  of  a  triangle  are  2,  3  and  4  inches.  Con- 
struct a  rectangle  equal  to  it,  having  one  side  of  2J 
inches.  Measuring  the  other  side  of  the  rectangle, 
calculate  approximately  the  area  of  the  rectangle,  i.e., 
of  the  triangle. 

The  sides  of  a  triangle  are  3  and  4  inches,  and  the 
included  angle  is  50°;  construct  a  rectangle  equal  to  it, 
one  of  whose  sides  is  2  inches.  Measuring  the  other 
side  of  the  rectangle  to  the  nearest  sixteenth  of  an 
inch,  calculate  approximately  the  area  of  the  rectangle, 
i.e.,  of  the  triangle. 

8.  If  we  wish  to  construct  a  rectangle  equal  in  area 
to  a  polygon,  and  thence,  if  necessary,  calculate  the 
area  of  the  polygon,  it  is  well  first  to  construct  a 
triangle  equal  to  the  polygon  by  the  following  method : 

Let  ABCD  be  a  quadrilateral 
whose  area  we  wish  to  calculate. 
Place  the  edge  of  the  parallel 
rulers  along  AC,  and  slide  one 
bar  out  until  the  edge    reaches     ^  C       t 

D,  and  mark  the  point  E  in  BC  produced.  AC  is 
parallel  to  DE,  and  therefore  the  triangle  ACE  is 
equal  to  the  triangle  ACD  (Ch.  VIII.,  3)  j  and  there- 
fore the  triangle  ABE  is  equal  to  the  quadrilateral 
ABCD. 

We  may  then  measure  the  perpendicular  height  of 
ABE  and  its  base  BE :  their  product  divided  by  2  gives 
the  area  of  ABE  (Ch.  VIII. ,  5),  and  therefore  of  ABCD. 


Areas  of  Parallelograms  and  Triangles.   63 

Suppose  we  wish  to  find  the  area  of  the  pentagon 
ABCDE.  Place  the  edge  of  the  parallel  rulers  along 
CE,  and  slide  one  bar  out 
until  the  edge  reaches  D,  /^vT^"^-^ 

and  mark  the  point  F  in 
BC  produced.  DF  is  par- 
allel to  CE,  and  therefore 
the  triangle  ECF  is  equal 
to  the  triangle  ECD.  Thus 
the  quadrilateral  ABFE  is  equal  to  the  pentagon 
ABCDE. 

Again,  place  the  edge  of  the  parallel  rulers  along 
AF,  and  slide  one  bar  out  until  the  edge  reaches  E, 
and  mark  the  point  6  in  BC  produced.  EG  is  parallel 
to  AF,  and  therefore  the  triangle  GAF  is  equal  to  the 
triangle  EAF  ;  and  therefore  the  triangle  ABG  is  equal 
to  the  quadrilateral  ABFE,  and  to  the  pentagon 
ABCDE. 

We  may  then  measure  the  perpendicular  height  of 
ABG  and  its  base  BG :  their  product  divided  by  2 
gives  the  area  of  ABG  and  therefore  of  ABCDE. 

In  the  preceding  figures  the  dotted  lines  need  not 
be  drawn.  The  point  E  in  the  former  figure,  and  the 
points  F  and  G  in  the  latter,  are  where  the  edge  of 
the  parallel  rulers  cuts  BC. 

Had  we  selected  AB  as  our  base,  instead  of  BC,  our 
resulting  triangle  would  have  had  a  different  height 
and  base,  but  would  necessarily  have  been  of  the 
same  area  as  ABE  or  ABG. 

The  sides  AB,  BC,  CD  of  a  quadrilateral  are  70,  60 
and  50  millimetres  j  the  angles  ABC,  BCD  are  70°  and 
60°.     Construct   a   triangle    equal   to   it   in   area,    and 


64  GrEOMETKY. 

tlieuce  calculate  its  area.  Here  use  BC  and  next  AB 
as  bases ;  and,  by  comparing  the  areas  of  the  result- 
ing triangles,  obtain  a  test  of  the  accuracy  of  your 
construction. 

Construct  several  quadrilaterals  and  pentagons,  and 
find  triangles  equal  to  them  in  area.  In  each  case 
construct  the  triangle  in  two  different  ways  (as  in 
the  preceding  example)  and,  by  comparing  the  areas 
of  such  triangles,  obtain  a  test  of  the  accuracy  of 
your  construction. 

In  all  cases  where  numerical  measurements  are 
made,  such  measurements  are  necessarily  approximate, 
and  therefore  in  examples  such  as  the  preceding  the 
areas  wiU  be  found  only  approximately.  Hence  where 
a  numerical  area  has  been  reached  by  two  different 
ways,  we   are   to   expect   only  approximate   agreement. 


CUapters  XIX.,  XX.,  and  XXI.,  relating  to  similar  triangles,  may 
now  be  taken  up  If  tlioiigtat  desirable. 

Exercises. 

1.  If  two  triangles  have  the  same  base  and  equal  areas,  what  rela- 
tion exists  between  their  altitudes  ? 

If  their  vertices  be  joined,  what   position   does  it  occupy  with 
respect  to  the  common  base  ? 

2.  If  D  and  E  be  the  middle  points  of  the  sides  AB,  AC  of  the 
triangle  ABC,  what  relation  exists  between  the  areas  of  the  triangles 
DBC,  EBC  ?  What  do  you  infer  as  to  the  position  of  DE  with  respect 
toBC? 

3.  Construct  a  quadrilateral,  and  bisect  the  sides.  What  positions 
do  the  lines  joining  the  bisections  of  adjacent  sides  occupy  with 
respect  to  the  diagonals  ? 

What  is  the  figure  formed  by  joining  in  succession  the  points  of 
bisection  ? 


Exercises.  65 

4.  Construct  a  quadrilateral,  and  bisect  the  sides.  How  do  the 
lines  joining  the  bisections  of  opposite  sides  divide  each  other  ?/  Give 
reason. 

5.  Two  sides  of  a  quadrilateral  are  parallel  and  of  lengths  2|  and 
3  inches.  The  distance  of  these  sides  apart  is  |  of  an  inch.  What  is 
the  area  of  the  quadrilateral  ?  (Join  two  opposite  corners,  and  find 
area  of  each  triangle.) 

6.  The  sides  of  a  rectangle  are  2  and  3  inches.  Find  by  geometri- 
cal construction  a  rectangle  equal  to  it  in  area,  one  of  whose  sides  is 
2J  inches.  Test  by  measurement  and  numerical  calculation  the 
accuracy  of  your  construction. 

7.  The  sides  of  a  triangle  are  3,  4^  and  5  inches.  Construct  a 
rectangle  equal  to  it  in  area  with  one  side  2^  inches.  Construct  also 
a  rectangle  equal  to  it  in  area,  one  of  whose  sides  is  3  inches. 

8.  The  base  of  a  triangle  is  70  millimetres,  and  the  angles  at  the 
base  30°  and  50°.  Construct  a  rectangle  equal  to  it  in  area,  one  of 
whose  sides  is  45  millimetres. 

9.  The  sides  of  a  rectangle  are  30  and  40  millimetres.  Construct  a 
parallelogram  equal  to  it  in  area,  one  of  whose  sides  is  30  millimetres, 
and  one  of  whose  angles  is  60°. 

10.  On  a  base  of  35  millimetres  construct  two  parallelograms  of 
equal  area,  one  having  a  side  of  55  millimetres  and  an  angle  of  75°, 
and  the  other  an  angle  of  120°. 

11.  The  sides  of  a  triangle  are  2  and  3  inches,  and  the  included 
angle  45°.  Construct  a  rectangle  equal  to  it  in  area,  one  of  whose 
sides  is  2^  inches. 

12.  In  the  previous  question,  construct  a  parallelogram  equal  to 
the  triangle,  with  one  of  its  angles  45°. 

13.  In  a  quadrilateral  ABCD,  AB=:35,  BC  =  45,  CD  =  55  milli- 
metres ;  ABC  =  60°,  BCD  =  75°.  Construct  a  triangle  and  also  a  rect- 
angle equal  to  it  in  area.  Hence  calculate  its  area,  approximately, 
in  square  millimetres. 

14.  A  quadrilateral  ABCD  has  AB  (2  in.)  and  CD  (3^  in.)  parallel, 
and  1^  in.  apart.  Construct  a  rectangle  equal  to  it  in  area,  one  of 
whose  sides  is  1^  in. 

15.  ABC  is  a  triangle,  and  D,  E  the  middle  points  of  AB,  AC. 
BE,  CD  intersect  in  O.  Join  AO,  and  show  that  the  triangles  OAB, 
OBC,  OCA  are  equal  in  area. 


66  GrEOMETRY. 

16.  In  the  previous  question,  if  F  be  the  middle  point  of  BC,  and 
OF  be  joined,  what  relation  holds  between  the  areas  of  the  six  tri- 
angles OAD,  ODB,  .  .  .  with  vertex  at  0  ? 

17.  In  the  same  question,  what  is  the  position  of  AO,  OF  with  re- 
spect to  each  other  ?     Test  and  give  reasons. 

18.  Construct  two  equal  triangles  on  the  same  base  and  on  opposite 
sides  of  it.  What  is  the  only  restriction  as  to  the  positions  of  their 
vertices  ?  If  the  vertices  be  joined,  how  is  the  joining  line  divided 
by  the  base,  or  base  produced  ? 

19.  From  any  point  in  an  equilateral  triangle  draw  perpendiculars 
to  the  sides.  What  relation  exists  between  their  sum  and  the  altitude 
of  the  triangle  ?     Give  reasons. 

20.  The  sides  of  a  right-angled  triangle  are  3,  4  and  5  inches.  If 
a  ^oint  within  the  triangle  be  1  inch  from  each  of  the  sides  contain- 
ing the  right  angle,  how  far  is  it  from  the  hypotenuse  ? 

21.  In  a  quadrilateral  ABCD,  AB  =  2,  BC  =  3,  and  CD=li  inches. 
ABC  =  35°,  BCD  =  100°.  Construct  a  triangle  and  also  a  rectangle 
equal  to  it  in  area.  Hence  calculate  the  area  of  ABCD,  approximate- 
ly, in  square  inches. 


CHAPTER  IX. 


Squares  on  Sides  of  a  Bigrht-angrled  Triangrle. 

1.  Let  the  angle  B  of  the 
triangle  ABC  be  90°,  Describe 
squares  on  the  sides  of  ABC, 
as  in  the  figure.  Draw  the 
lines  AG,  EF,  CH  parallel  to 
BC;  and  the  lines  DK,  EH 
parallel  to  AB. 

Then  measurement  (with 
dividers  for  lines,  and  bevel 
for  angles)  will  show  that  the 
triangle  AGD  is  in  all  respects 

equal  to  the  triangle  ABC ;  and  cutting  out  the  triangle 
AGD,  it  may  be  turned  about  A,  in  the  direction  indi- 
cated by  the  arrow  head,  into  the  position  ABC. 
Measurement  will  also  show  that  the  triangle  EFD  is 
in  all  respects  equal  to  the  triangle  EHC ;  and  cutting 
out  the  triangle  EFD,  it  may  be  turned  about  E,  in  the 
direction  indicated  by  the  arrow  head,  into  the  position 
EHC.  We  thus  have  the  square  on  AC  converted  into 
ABKG  and  FKHE,  which  will  be  found  to  bo  the  squares 
on  AB  and  BC. 

Repeat    the    same    construction,   measurements,    and 
superposition  in  the  case  of  the  following  triangles: 

AB  =  35,  BC  =  50  millimetres  5  ABC  =  90°. 
AB  =  1J  in.,  BC  =  2J  in.  5  ABC  =  90°. 
AB  =  2  in.  J  ABC  =  90°,  BAC  =  60°. 
67 


68 


GrEOMETRY. 


The  result  of  these  observations  may  be  stated  thus : 
In  any  right-angled  triangle  the  square  which 
is  described  on  the  side  subtending  the  right 
angle,  is  equal  to  the  sum  of  the  squares  de- 
scribed on  the  sides  containing  the  right  angle. 

Two  sides  of  a  right-angled  triangle  about  the  right 
angle,  are  3  and  4.  What  is  the  length  of  the  third 
side? 

If  a  string  or  rope  of  length  12  be  broken  into 
lengths  dj^A  and  5,  and  these  be  formed  into  a  triangle, 
such  triangle  is  right-angled. 

If  the  lengths  of  the  pieces  of  rope  be  30,  40  and 
50,  the  triangle  formed  with  them  is  also  right-angled. 

2.  A  square  may  be  constructed  equal  in  area  to  any 
rectangle,  as  follows: 

Let  ABCD  be  the  rectangle. 
Make  DE  equal  to  DC,  and 
find  F  the  middle  point  of 
AE.  Describe  the  semicircle, 
and  produce  CD  to  G.  Then 
the  square  on  DG  is  equal 
to  the  rectangle  ABCD. 

For,  describe  the  square  DGLK  on  DG,  and  let  LK 
and  BC  meet  in  H.  Then,  if  the  figure  has  been 
accurately  constructed,  on  producing  the  lines  LG,  HD 
and  BA,  they  will  be  found  to  all  pass  through  one 
point,  M.  Hence  (Ch.  VIII.,  6)  the  square  GDKL  is 
equal  to  the  rectangle  ABCD. 

In  the  succeeding  constructions  it  is  of  course  abso- 
lutely necessary  that  the  three  lines  corresponding  to 
LG,  HD  and  BA  pass  through  the  same  point  (M). 


Exercises.  69 

Describe  the  rectangle  whose  sides  are  40  and  90 
millimetres.  Construct,  as  above,  the  square  equal  to 
it.  Measure  in  millimetres  the  side  of  the  square,  and 
thence  verify  the  accuracy  of  your  construction. 

Proceed  similarly  with  the  rectangle  whose  sides  are 
1  and  4  inches. 

Also  with  the  rectangle  whose  sides  are  9  and  16 
sixteenths  of  an  inch. 

^Iso  with  the  rectangle  whose   sides  are  18  and  32 
sixteenths  of  an  inch.     The  sides  of  this  rectangle  are 
twice  those  of  the  former :    note  the  numbei*  qf  times 
its  area  is  greater  than  that   of  the   former;   note~th^ 
same  with  respect  to  the  resulting  squares. 

ABC  is  a  right-angled  triangle, 
ABC  being  the  right  angle-  and 
BD  is  perpendicular  to  AC. 

Construct  the  rectangle  whose 
sides   are  CA,   AD ;    by  the  pre- 
ceding method   construct  the   square   equal  to  it,   and 
show  that  it  is  the  square  on  AB. 

Similarly  by  construction  show  that  the  rectangle 
contained  by  AC,  CD  is  equal  to  the  square  on  BC. 

Also  that  the  rectangle  contained  by  AD,  DC  is  equal 
to  the  square  on  BD. 

Exercises. 

1.  Three  straight  lines,  of  lengths  3,  4,  5,  forming  a  right-angled 
triangle,  what  sort  of  triangle  is  formed  by  lines  of  lengths  6,  8,  10, 
or  9,  12,  15,  or  12,  16,  20,  etc.  ? 

2.  Construct  triangles  with  sides  as  follows  :  3,  4,  5  inches  ;  30,  40, 
50  millimetres  ;  36,  48,  60  millimetres  ;  3|,  5,  6^  inches.  Compare 
the  angles  of  these  triangles,  and  state  the  result  of  such  comparison. 
What  relation  do  the  sides  of  one  triangle  bear  to  the  sides  of 
another  ? 


70  Geometky. 

3.  Given 

(2n+  1)'  +  i2n^+2ny  =  {2n^  +  2^+1)2 

by  assigning  to  n  in  succession  the  values  1,  2,  3,  ...  ,  form  a  series 
of  whole  numbers,  in  groups  of  three,  such  that  each  group  gives  the 
lengths  of  the  sides  of  a  right-angled  triangle. 

4.  The  side  of  an  equilateral  triangle  is  2.  What  is  the  length  of 
the  perpendicular  from  any  angle  on  the  opposite  side  ? 

5.  Draw  two  lines  CA,  CB  at  right  angles  to  each  other  and  each 
of  length  one  inch.     What  is  the  area  of  the  square  on  AB  ? 

6.  In  the  iSgure  of  the  preceding  question,  draw  AD  ( =  1  in. )  per- 
pendicular to  AB.     What  is  the  area  of  the  square  on  DB  ? 

7.  In  the  same  figure  draw  DE  (=  1  in.)  perpendicular  to  DB. 
What  is  the  area  of  the  square  on  EB  ?  Test  by  measuring  the 
length  of  EB. 

8.  Construct  a  square  which  shall  contain  13  square  inches. 

9.  Test  the  accuracy  of  the  construction  in  the  preceding  question 
by  drawing,  at  right  angles  to  the  side  of  the  square,  a  line  equal  to 
the  side  of  a  square  containing  3  square  inches,  joining  the  ends  of 
the  lines,  and  measuring  the  hypotenuse  of  the  right-angled  triangle 
so  obtained. 

10.  Describe  squares  on  the  sides  of  a  right-angled  triangle.  Con- 
struct another  triangle  with  sides  equal  to  the  diagonals  of  these 
squares.     What  is  this  latter  triangle  ? 

11.  In  the  preceding  question  by  what  multiplier  can  you  obtain 
the  sides  of  one  triangle  from  those  of  the  other  ? 

Compare  the  angles  of  the  two  triangles  and  state  the  result  of 
such  comparison. 

12.  Describe  a  triangle  such  that  the  square  on  one  side  is 
greater  than  the  sum  of  the  squares  on  the  two  other  sides,  say 
with  sides  of  2,  3  and  4  inches.  What  relation  does  the  angle 
opposite  the  greatest  side  bear  to  a  right  angle  ?  Measure  it  with 
protractor. 

13.  Construct  a  triangle  with  sides  of  30,  40  and  55  millimetres 
(55^>30^  +40^).  What  sort  of  angle  is  that  opposite  the  greatest 
side? 

14.  Describe  a  triangle  such  that  the  square  on  one  side  is  less 
than  the  sum  of  the  squares  on  the  two  other  sides,  say  with  sides  of 


Exercises.  71 

40,  60  and  65  millimetres.     What  relation   does  the  angle  opposite 
the  greatest  side  bear  to  a  right  angle  ? 

15.  Construct  a  triangle  with  sides  of  2,  3  and  3|f  inches 
(3.5^<2^+3^).     What  is  the  angle  opposite  the  side  of  3^  inches  ? 

16.  Construct  any  quadrilateral  with  its  diagonals  at  right  angles 
to  each  other.  Show  that  the  sum  of  the  squares  on  two  opposite 
sides  is  equal  to  the  sum  of  the  squares  on  the  other  two  sides. 

17.  Describe  a  square  ABCD,  and  in  the  sides  take  points  E,  F,  G, 
H,  such  that  AE  =  BF  =  CG  =  DH.  What  is  the  figure  EFGH.  Apply 
tests.     Give  reasons. 

18.  Two  squares  being  given,  say  of  9  and  16  square  inches,  show 
how  to  draw  a  line  the  square  on  which  shall  be  equal  to  the  differ- 
ence of  these  given  squares. 

19.  ABC,  A'B'C  are  right-angled  triangles  with  the  hypotenuses 
AB,  A'B'  equal,  and  also  the  sides  BC,  B'C  equal.  Show  that 
the  remaining  sides  AC,  A'C  are  equal. 

20.  The  sides  of  a  triangle  are  1|,  2,  2^  inches.  Construct  a 
square  equal  to  it. 

21.  The  side  of  an  equilateral  triangle  is  2  inches.  Construct  a 
square  equal  to  it. 

22.  The  sides  of  a  rectangle  are  24  and  54  sixteenths  of  an  inch. 
Construct  a  square  equal  to  it.  Measure  the  side  of  the  square,  and 
thence  verify  the  accuracy  of  your  construction. 

23.  If  a  right-angled  triangle  have  one  of  the  acute  angles  double 
the  other,  divide  it  into  two  triangles,  one  equilateral  and  the  other 
isosceles. 

24.  Bisect  the  hypotenuse  of  a  right-angled  triangle.  What  rela- 
tion between  the  distances  of  the  point  so  obtained  from  the  three 
angles  ? 

25.  ABC  is  a  right-angled  triangle,  and  CD  is  drawn  perpendicular 
to  the  hypotenuse.  Examine  the  relations  between  the  angles  of  the 
three  triangles  ABC,  ACD,  BCD.     Give  reasons. 


CHAPTER  X. 


The  Circle.     Its  Syminetry.     Tang-ents. 

Centre. 


Findinsr  of 


1.  The  fundamental  quality  of 
the  circle,  next  to  the  equality  of 
its  radii,  is  its  symmetry. 

In  the  first  place,  every  line 
drawn  through  the  centre  from 
circumference  to  circumference 
(i.e. J  every  diameter)  is  bisected 
at  the  centre.  This  is  called 
central  symmetry. 

In  the  second  place,  every 
chord  drawn  at  right  angles  to 
a  diameter  is  bisected  by  that 
diameter.  This  is  called  axial 
sym.metry.  Thus  the  chord 
EFG  being  perpendicular  to  OA, 
the  parts  EF,  FG  are  equal. 
Measurement  will  establish  the 
equality  of  these  parts.  Or  we 
may  prove  it  thus : 

The  rt.    zl^es  at  F  are  equal. 

Because  OE  =  OG,  .-.   L   OEF=ZOGF. 

Hence   Z  ^^^  at  0  are  equal. 

Also  sides  EO,  OF  =  sides  GO,  OF. 

.-.  (Ch.  III.,  2)  EF  =  FG. 

And  hence  all  chords  perpendicular  to  a  diameter   are 

bisected  by  it. 

2.  As  the  chord  BCD  moves  parallel  to  itself  down 
to. A,  since  the  parts  on  each  side  of  the  diameter  are 
always  equal,  when  one  part  vanishes,  the  other  vanishes 

72 


B^-^ 

C^-^D 

/ 

X 

/ 

F               \ 

^ 

0 

The  Cikcle,  Its  Symmetry,  Tangents,  Etc.  73 


also.  Thus  the  line  TAP,  through  A  parallel  to  BCD, 
while  it  touches  the  circle,  does  not  cut  it.  Such 
a  line  (TAP)  is  called  the  tang^ent  to  the  circle  at  A. 
That  is  to  say,  a  tangent  is  a  line  drawn  through 
the  extremity  of  a  diameter,  and  at  right  angles 
to  it. 

The  tangent  is  evidently  a  straight  line  which  meets 
the  circle,  but  does  not  cut  it :  this  is  sometimes  given 
as  the  definition  of  a  tangent. 

3.  Since  a  diameter  bisects  every  chord  to  which  it 
is  at  right  angles,  therefore  a  line  drawn  through  the 
bisection  of  a  chord  and  at  right  angles  to  it,  must 
be  a  diameter.  Hence  if  the  centre  of  any  circle  be 
not  indicated,  we  may  reach  it  by  the  following  con- 
struction : 

Draw  any  chord  AB.  Bisect 
it  at  C.  Draw  DCE  perpendicu- 
lar to  AB.  DE  must  pass 
through  the  centre.  Hence,  bi- 
secting DE  at  F,  F  must  be 
the  centre  of  the  circle. 

We  may  describe  circles  with- 
out marking  their  centres  by 
placing  a  piece  of  thin  wood 
or  cardboard  under  the  station- 
ary point  of  the  compasses,  removing  this  piece  of  wood 
or  cardboard  when  the  circle  is  described. 

Circles  being  thus  described,  or  being  obtained  by 
marking  with  the  pencil  about  a  round  object  placed 
on  the  paper  (coin,  bottom  of  ink  bottle,  plate,  &c.), 
attempts  should  be  made  to  locate  the  centre  by  the 
eye's  judgment.  We  may  afterwards  test  the  correct- 
ness  of   this   by   making  the   preceding    construction, 


74  Geometry. 

and  finally  test  the  accuracy  of  the  construction  by 
trying  with  such  centre  to  reproduce  the  circle  by  using 
the  compasses. 

It  will  be  found,  of  course,  that  the  greater  the 
circle,  the  greater  will  be  the  difficulty  of  locating, 
with  the  eye's  judgment,  the  position  of  the  centre. 
The  same  difficulty  occurs  in  locating  the  bisection  of 
a  straight  line  with  the  eye. 

4.  If  only  an  arc  of  the 
circle  be  given,  we  may  find 
the  centre,  and  complete  the 
circle,  as  follows : 

Draw  two  chords  AB  and 
CD ;  find  their  middle  points 
E  and  F  ;  through  these 
middle  points  draw  perpendiculars  EG  and  FH.  The 
centre  of  the  circle  must  lie  on  each  of  the  lines  EG 
and  FH  (Ch.  X.,  3),  and  therefore  must  be  at  0. 

Arcs  of  circles  should  be  described  without  marking 
the  centres,  by  the  method  suggested  in  §  3.  The  posi- 
tions of  the  centres  should  then  be  judged  with  the 
eye;  afterwards  constructed  for,  and  the  accuracy  of 
the  construction  tested  by  attempting,  with  the  com- 
passes, to  describe  the  arc  with  the  centre  so  obtained. 

5.  If  any  line  AB  be 
bisected  at  C,  and  CD  be 
drawn  perpendicular  to 
it,  then  all  points  in  CD 
are  equally  distant  from 
A  and  B.  Hence  if  we 
place  the  sharp  point  of 
the  compasses  at  any 
point   on  CD,    and  the 


The  Cikcle,  Its  Symmetky,  Tangents,  Etc.  75 

pencil  end  at  A,  and  describe  a  circle,  it  will  also  pass 
through  B.  We  thus  get  an  unlimited  number  of 
circles  through  A  and  B,  all  of  which  have  their  centres 
at  different  points  on  CD. 

Draw  a  line  AB  of  50  millimetres,  and  describe  circles 
passing  through  A  and  B,  with  radii  30,  40,  50  and  60 
millimetres. 

6.  We  can  readily  ob- 
tain a  method  for  de- 
scribing a  circle  to  pass 
through  any  three  points : 

Let  A,  B,  C  be  the  three 
points.  Draw  DO  from  the 
middle  point  of  AB  at  right 
angles  to  it ;  and  draw  EO 
from  the  middle  point  of  BC  at  right  angles  to  it. 
Then  all  points  in  DO  are  equally  distant  from  A  and 
B ;  and  all  points  in  EO  are  equally  distant  from  B  and 
C.  Hence  0  is  equally  distant  from  A,  B  and  C;  and 
placing  the  sharp  point  of  the  compasses  at  0  and 
the  pencil  end  at  A,  and  describing  a  circle,  it  will 
pass  through  B  and  C,  if  the  construction  has  been 
accurate. 

AB  is  1  inch,  BC  is  2  inches,  and  angle  ABC  is  120°. 
Describe  a  circle  to  pass  through  A,  B  and  C. 

AB  is  40  and  BC  60  millimetres,  and  the  angle  ABC 
is  75°.     Describe  a  circle  to  pass  through  A,  B  and  C. 

AB  is  li  and  BC  2 J  inches,  and  the  angle  ABC  is 
90°.  Describe  a  circle  to  pass  through  A,  B  and  C. 
Show  that  its  centre  bisects  AC. 

Mark  sets  of  three  points  in  various  positions  with 
respect  to  one  another,  and  describe  a  circle  to  pass 
through  each  set. 


76  Geometry. 


Exercises. 

1.  Describe  a  circle;  draw  in  it  any  chord;  join  the  centre  to  the 
extremities  of  the  chord  ;  and  drop  a  perpendicular  from  centre  on 
chord. 

What  is  the  relation  between  the  angles  the  radii  make  wi^h  the 
chord?  What  between  the  angles  the  radii  make  with  the  perpen- 
dicular? What  between  the  segments  of  the  chord  made  by  the  foot 
of  the  perpendicular  ? 

2.  With  the  bevel  or  ^protractor  construct  two  equal  angles  at  the 
centre  of  a  circle,  and  draw  the  chords  which  subtend  these  angles. 
What  is  the  relation  between  these  chords  ?  Apply  test.  Give 
reasons. 

3.  Describe  a  circle,  and  with  dividers  and  ruler  place  two  equal 
chords  in  it.  Join  the  ends  of  the  chords  to  the  centre.  What  is  the 
relation  between  the  angles  these  equal  chords  subtend  at  the  centre  ? 
Apply  test.     Give  reasons. 

4.  As  in  the  previous  question,  in  a  circle  place  two  equal  chords, 
and  from  the  centre  drop  perpendiculars  on  them.  What  is  the  re- 
lation between  these  perpendiculars  ?     Apply  test.     Give  reasons. 

5.  Describe  a  circle  of  radius  3  inches,  from  the  centre  draw  two 
equal  lines  of  length  2  inches,  and  through  the  extremity  of  each 
draw  a  line  at  right  angles  to  it,  so  obtaining  two  chords  at  equal 
distances  from  the  centre.  What  is  the  relation  between  the  lengths 
of  these  chords  ?     Apply  test.     Give  reasons. 

6.  The  sides  of  a  triangle  are  2^,  3  and  3^  inches.  Describe  a 
circle  passing  through  the  angular  points. 

7.  The  sides  of  a  triangle  are  2,  3  and  4  inches.  Describe  a  circle 
passing  through  the  angular  points. 

8.  The  sides  of  a  triangle  are  3,  4  and  5  inches.  Describe  a  circle 
passing  through  the  angular  points. 

9.  Two  chords  of  a  circle  with  one  end  of  each  common,  are  of 
lengths  2  and  3  inches,  and  make  an  angle  of  60°  with  each  other. 
Describe  the  circle. 

10.  Two  chords  of  a  circle  make  angles  of  50°  and  60°  with  a  third 
chord  whose  length  as  2^  inches,  and  are  inclined  towards  one  another. 
Describe  the  circle 


■    -  EXEKCISES.  77 

11.  The  sides  of  a  rectangle  are  40  and  60  millimetres.  Describe  a 
circle  passing  through  all  the  angular  points. 

12.  Describe  a  parallelogram  ABCD,  not  beixig  a  rectangle.  Can  a 
circle  be  described  passing  through  its  angular  points  ?  (Every  circle 
through  A  and  B  has  its  centre  on  the  line  which  bisects  AB  at  right 
angles.  ^ 

13.  The  diameter  of  a  circle  is  30  inches,  and  a  chord  is  24  inches. 
How  far  is  the  chord  from  the  centre  ? 

14.  The  radius  of  a  circle  is  3^  inches.  What  is  the  length  of  a 
chord  whose  distance  from  the  centre  is  1 J  inches  ? 

15.  The  equal  sides  AB,  AC,  of  an  isosceles  triangle  ABC,  are  50 
millimetres,  and  they  contain  an  angle  of  45°.  A  circle  with  centre  A, 
and  radius  70  millimetres,  cuts  BC  produced  in  D  and  E.  What  is 
the  relation  between  the  lengths  of  DB  and  CE  ?  Apply  test.  Give 
reasons. 

16.  Describe  a  circle  ;  draw  a  diameter,  producing  it ;  and  from  a 
point  A  in  the  produced  diameter  draw  two  lines  on  opposite  sides  of 
it,  making  equal  angles  with  it.  What  do  you  observe  as  to  the 
lengths  of  the  segments  of  these  lines  between  A  and  the  points  of 
section  by  the  circle  ?  What  as  to  the  parts  within  the  circle? 
Apply  tests.     Give  reasons. 

17.  The  same  question  as  the  preceding,  but  with  A  within  the 
circle. 

18.  Construct  two  intersecting  circles,  join  their  centres,  and 
through  either  of  the  points  of  intersection,  draw  a  line  parallel  to  the 
line  joining  centres,  and  terminated  by  the  circumferences.  What 
relation  in  length  between  the  second  line  drawn  and  the  line  joining 
the  centres  ?     Apply  test.     Give  reasons. 

19.  AB,  CD  are  two  parallel  chords  in  a  circle.  What  relation 
exists  between  the  lengths  of  the  chords  AC,  BD  ?  Apply  test.  Give 
reasons. 

20.  In  the  previous  question  prove  angle  ABD  =  angle  BAC  :  also 
anffle  ACD  =  BDC  :  also  chord  AD = chord  BC. 


CHAPTER  XI. 

Tangrents  to  Circles,  and  Circles  Touching-  One 
Another. 

1.  To  draw  the  tangent 
at  any  point  A  on  the  cir- 
cumference of  a  circle, 
draw  the  diameter  through 
A,  and  draw  at  A  the  per- 
pendicular to  this  diameter. 
The  perpendicular  is  a  tan- 
gent to  the  circle  (Ch.  X., 
2). 

Evidently  the  tangents  at  opposite  ends  of  a  diameter 
are  parallel  to  one  another. 

Construct  a  circle  of  radius  55  millimetres.  Draw 
radii  at  intervals  of  30°,  and  draw  the  tangents  at  the 
ends  of  these  radii,  producing  each  both  ways  until 
it  meets  the  adjacent  tangents. 

Construct  a  circle  of  radius  49  millimetres.  Draw 
radii  at  intervals  of  45°,  and  draw  tangents  at  the 
ends  of  these  radii,  producing  each  both  ways  until 
it  meets  the  adjacent  tangents. 

Construct  a  circle  of  radius  l-f^-  in.  Draw  radii  at 
intervals  of  72°,  and  draw  tangents  at  the  ends  of 
these  radii,  producing  each  both  ways  until  it  meets 
the  adjacent  tangents. 

In   each   of  the   three   preceding   constructions,  the 

78 


Tangents  to  Circles,  Etc. 


79 


resulting  figure  about  the  circle  should  have  equal 
sides  and  equal  angles.  The  equality  of  the  sides 
(measured  with  the  dividers)  and  the  equality  of  the 
angles  (measured  with  the  bevel)  may  be  regarded  as 
a  test  of  the  accuracy  of  the  construction. 

Any  two  diameters  in  a  circle  are  drawn,  inclined 
at  an  angle  of,  say,  30°  to  each  other,  and  tangents  at 
the  ends  of  these  diameters  are  constructed.  What 
quadrilateral  figure  about  the  circle  do  the  tangents 
form?     Measure  its  sides. 

2.  From  a  point  without  a  cir- 
cle, evidently  two  tangents  can 
be  drawn  to  the  circle.  To  draw 
those  from  A  to  the  circle  FBG : 

Join  AC,  cutting  the  circle  in 
B.  Describe  a  second  circle  DAE, 
with  centre  C  and  radius  CA. 
Draw  DBE  perpendicular  to  CB. 
Join  CD  and  CE,  cutting  the  small 

circle  in  F  and  G.  Then  AF  and  AG  are  the  tangents 
from  A. 

For,  the  triangles  ACF  and  DCB  are  equal.  But  the 
angle  CBD  is  a  right  angle  j  therefore  the  angle  CFA 
is  a  right  angle,  and  AF  is  a  tangent  to  the  circle 
(Ch.  X.,  2).  In  the  same  way  we  may  prove  that  AG 
is  a  tangent. 

Symmetry  suggests  that  the  tangents  AF,  AG  are 
equal  in  length,  and  that  they  make  equal  angles  with 
AC.  The  truth  of  this  may  be  tested  by  measure- 
ment. It  may  also  be  proved  as  follows:  Because 
CDE  is  an  isosceles  triangle,  and  the  angles  at  B  right 


80' 


Geometky. 


angles,  therefore  the  triangles  CDB,  CEB  are  equal  in 
all  respects.  But  the  triangle  CAF  is  equal  in  all 
respects  to  CDB ;  and  the  triangle  CAG  is  equal  in  all 
respects  to  the  triangle  CEB.  Therefore  the  triangles 
CAF  and  CAG  are  equal  in  all  respects.  Hence  AF,  AG 
are  equal,  and  the  angles  at  A  are  equal. 

In  practice,  an  easy  way  to 
draw  a  tangent  from  any 
point  A,  outside  the  circle, 
is  as  follows:  Place  the  set- 
square  so  that  one  of  its 
sides  passes  through  A  and 
the  other  through  C,  the  cen- 
tre of  the  circle.  Then  so 
adjust  the  instrument  that 
the  right  angle  rests  on  the  circumference  at,  say,  B. 
AB,  a  tangent  through  A,  may  then  be  drawn. 

Construct  a  circle  of  radius  IJ  in.,  and  draw  any 
line  through  its  centre.  From  points  on  this  line  at 
distances  from  the  centre  2,  2 J,  3  in.,  draw  tangents  to 
the  circle. 


3.  Let  a  circle  be  described 
with  centre  A,  and  the  tan- 
gent at  any  point  C  be  drawnj 
and  let,  with  centre  B,  on 
AC,  and  radius  BC,  another 
circle  be  drawn.  Then  both 
circles  have  CD  for  tangent. 
Both  touching  the  same  line 
at  the  same  point,  they  are 
said  to  touch  one  another, — in  this  case  internally. 


Tangents  to  Ciecles,  Etc.  81 

Let  a  circle  be  described  with 
centre  A,  and  the  tangent  at  any 
point  C  be  drawn  j  and  let,  with 
centre  B,  in  AC  produced,  and  ra- 
dius BC,  another  circle  be  described. 
Then  both  circles  have  CD  for 
tangent.  Both  touching  the  same 
line  at  the  same  point,  they  are 
said  to  touch  one  another, — in  this  case  externally. 

Evidently,  whether  circles  touch  internally  or  exter- 
nally, the  straight  line  joining  their  centres  passes 
through  the  point  of  contact. 

Describe  circles  of  radii  34  and  56  millimetres  to 
touch  (1)  externally,  (2)  internally. 

Construct  a  series'  of  circles  of  radii  20,  17,  14,  11, 
.  .  .  millimetres,  their  centres  being  in  the  same  straight 
line,  and  each  circle  touching  the  preceding  (and  suc- 
ceeding) externally. 

Describe  circles  of  radii  as  in  preceding,  but  each 
circle  touching  the  others  at  the  same  point,  internally. 

Two  circles  of  radii  30  and  40  millimetres  touch  one 
another  externally.  Describe  a  circle  of  radius  20,  to 
touch  both  of  them  externally.  (This  involves  the 
construction  of  a  triangle  with  sides  70,  60  and  50 
millimetres.) 

Make  the  same  construction  as  in  the  preceding 
question,  when  the  first  two  circles  have  radii  25  and 
35,  and  the  third  a  radius  of  15  millimetres. 

The  sides  of  a  triangle  are  75,  60  and  45  milli- 
metres. "With  the  angular  points  of  this  triangle  as 
centres,  describe  three  circles  with  radii  15,  30  and  45 
millimetres,  so  that  each  may  touch  the  other  two. 


82  GrEOMETRY. 

When  the  sides  of  the  triangle  are  100,  75  and  65 
millimetres^  discover  the  circles  whose  radii  are  such 
that  in  like  manner  each  will  touch  the  other  two,  the 
angular  points  of  the  triangle  being  centres  of  the 
circles. 

Exercises. 

1.  Describe  a  circle  of  radius  40  millimetres ;  draw  two  diameters 
at  right  angles  to  one  another ;  and  draw  tangents  at  ends  of  the 
diameters,  and  produce  them  so  that  they  intersect.  What  do  you 
observe  as  to  lengths  of  tangents  ?  What  angles  do  they  make  with 
one  another  ?    Apply  tests  with  dividers  and  set-square. 

2.  Describe  a  circle  of  radius  1^  in.  ;  draw  diameters  at  intervals 
of  60° ;  and  draw  tangents  at  ends  of  diameters.  What  do  you 
observe  as  to  lengths  of  tangents  ?  What  angles  do  they  make  with 
one  another  ?     Apply  tests. 

3.  Describe  a  circle  of  radius  IJ  in.  ;  draw  any  line  in  plane  of 
paper ;  draw  a  tangent  parallel  to  this  line.  (From  centre  drop 
perpendicular  on  line,  and  at  point  of  intersection  with  circle  draw 
tangent. ) 

4.  Describe  a  circle  of  radius  35  millimetres ;  draw  any  line  in 
plane  of  paper  ;  draw  a  tangent  to  circle  which  shall  be  perpendicular 
to  this  line. 

5.  Draw  any  line  and  draw  circles  of  radii  1,  IJ  and  2  inches, 
touching  the  line  at  any  points. 

6.  Describe  two  circles  of  radii  1  inch  and  2J  inches,  so  as  to  touch 
any  line  at  points  3  inches  apart.     Do  the  circles  touch  one  another  ? 

7.  A  tangent  of  length  4  inches  is  drawn  from  a  point  to  a  circle  of 
radius  3  inches.     How  far  is  the  point  from  the  centre  of  the  circle? 

8.  A  tangent  is  drawn  to  a  circle  of  radius  1  inch,  and  another 
circle,  concentric  with  the  former,  is  described  of  radius  2  inches. 
What  is  the  length  of  the  tangent  between  the  point  where  it  is 
intercepted  by  the  second  circle  and  the  point  of  contact?  What 
angle  does  the  intercepted  portion  of  the  tangent  subtend  at  the 
common  centre  ? 


Exercises.  83 

9.  A  circle  has  a  radius  of  30  millimetres,  and  a  tangent  of  length 
40  millimetres  is  drawn  to  it.  What  line  (curved)  represents  all  the 
points,  outside  the  circle,  from  which  this  tangent  may  be  drawn? 

10.  From  four  points,  equidistant  from  one  another,  on  a  circle  of 
radius  2  inches,  draw  tangents  to  a  concentric  circle  of  radius  1  inch. 

11.  Describe  two  circles  of  radii  1  and  IJ  inches,  to  touch  one 
another ;  and  describe  a  circle  of  radius  2^  inches  to  touch  both,  and 
contain  both. 

12.  The  preceding  problem  with  each  circle  external  to  the  other 
two. 

13.  Describe  three  circles  of  radii  2^,  3  and  3^  inches,  so  that  each 
may  touch  the  other  two. 

14.  Describe  two  concentric  circles  of  radii  1  and  3  inches,  and 
describe  a  number  of  circles  touching  both  of  them. 

15.  Two  circles  touch  internally  at  A,  and  ABC  is  drawn  to  meet 
the  circles  at  B  and  C.  What  is  the  position  of  radii  to  B  and  C  with 
respect  to  each  other  ?     Apply  test.     Give  reasons. 

16.  Two  circles  touch  externally  at  A,  and  ABC  is  drawn  to  meet 
the  circles  at  B  and  C.  What  is  the  position  of  radii  to  B  and  C  with 
respect  to  each  other  ?     Apply  test.     Give  reasons. 

17.  OA,  OB  are  drawn  through  the  centre  of  a  circle  at  right  angles 
to  each  other,  and  a  tangent  to  the  circle  meets  these  lines  at  A  and 
B.  Two  other  tangents  are  drawn  to  the  circle  from  A  and  B.  What 
is  the  position  of  these  latter  tangents  with  respect  to  each  other  ? 
Apply  test.     Give  reasons. 

18.  Draw  two  tangents  to  a  circle  from  an  external  point,  and  join 
the  points  of  contact  What  is  the  relation  between  the  angles  this 
*'  chord  of  contact "  makes  with  the  tangents  ?  Apply  test.  Give 
reasons. 

19.  Two  circles  touch  externally  and  parallel  diameters  are  drawn. 
Lines  are  drawn  from  opposite  ends  of  these  diameters  to  the  point 
of  contact :  what  position  do  they  occupy  with  respect  to  each 
other  ? 

20.  Two  circles  touch  internally  and  parallel  diameters  are  drawn. 
Lines  are  drawn  from  corresponding  ends  of  these  diameters  to  the 
point  of  contact :  what  position  do  they  occupy  with  respect  to  each 
other  ? 


84  Geometry. 

21.  Describe  two  circles  with  radii  1^  in.  and  ^in.,  respectively, 
their  centres  being  3  in.  apart.  Concentric  with  the  larger,  describe 
a  third  circle  of  radius  f  in.  (I4-2);  and  from  the  centre  of  the 
smallest  circle  draw  a  tangent  to  this  third  circle.  Draw  a  line 
parallel  to  this  tangent,  and  at  distance  |  in.  from  it.  What  is  this 
last  line  with  respect  to  the  first  two  circles  ?     Apply  tests. 

22.  Describe  two  circles  with  radii  1^  in.  and  ^  in. ,  respectively, 
their  centres  being  3  in.  apart.  Concentric  with  the  larger  circle, 
describe  a  third  circle  of  radius  If  in,  (1|  +  2)  >  3-"^  from  the  centre  of 
the  smallest  circle  draw  a  tangent  to  this  third  circle.  Draw  a  line 
parallel  to  this  tangent,  and  at  distance  J  in.  from  it.  What  is  this 
last  line  with  respect  to  the  first  two  circles  ?     Apply  tests. 


CHAPTER  XII. 

Angrles  in  a  Circle. 

1.  The  angles  ACB,  ADB  stand  on  the  same  arc  AB, 
the  one  being  at  the  centre  and  the  other  at  the  cir- 
cumference. 

■■  0  -I-  /?'. 


Meastire  the  number  of  degrees  in  each,  and  com- 
pare these  numbers. 

Make  the  same  constructions  in  the  case  of  two  or 
three  other  circles,  and  repeat  the  measurements  and 
comparison. 

What  is  your  conclusion  as  to  the  size  of  the  angle 
at  the  centre,  compared  with  the  size  of  the  angle  at 
the  circumference? 

85 


86 


GrEOMETBY. 


The  relation  between  these  angles  may  be  reasoned 
out  as  follows : 

CAD  is  an  isosceles  triangle* 
and  therefore  the  angles  CAD,  CD  A 
are  equal.  Hence  the  exterior 
angle  ACE,  which  is  equal  to 
their  sum  (Ch.  V.,  1)^  must  be 
twice  ADC.  Similarly  BCE  is 
twice  BDC.  Therefore  the  sum 
(or  difference,  see  second  figure) 
ACB  is  twice.  ADB. 

That  is,  the  angle  at  the 
centre  of  a  circle  is  double 
the  angle  at  the  circumfer- 
ence which  stands  upon  the 
same  arc  (here  AB). 

The  truth  of  this  should  be 
tested  by  describing  a  number  of  circles,  constructing, 
in  each  case,  an  angle  at  the  centre  and  another  at 
the  circumference  on  the  same  arc,  and  using  the  pro- 
tractor to  determine  the  magnitudes  of  these  angles. 

2.  Construct  such  a  figure 
as  the  *  annexed,  where  an 
angle  ACB  at  the  centre,  and  a 
number  of  angles  ADB,  AEB, 
....  at  the  circumference, 
stand  on  the  same  arc  AB. 
Then,  adjusting  the  bevel  to 
the  angles  at  the  circumfer- 
ence, compare  their  magni- 
tudes. The  result  of  such  a 
comparison  might  have  been  anticipated,  since  each  of 


Angles  in  a  Cikcle.  87 

the  angles  at  the  circumference  is  half  the  same  angle, 
ACB,  at  the  centre. 

Hence  angles  described  in  the  same  segment 
of  a  circle,  i.e.,  angles  standing  on  the  same 
arc  of  a  circle,  being  on  the  circumference, 
are  equal  to  one  another. 

Using  the  bevel,  construct  a  number  of  angles  as  in 
the  annexed  figure,  all  of  the  same  magnitude,  and 
with  the  sides  of  each  passing 
through  the  points  A  and  B.  Then 
taking  any  three  of  the  angular 
points,  and,  by  the  method  of 
Ch.  X.,  6,  constructing  for  the 
circle .  through  these  three  points, 
show,  by  describing  the  circle,  that 
it  passes  through  the  other  angu- 
lar points,  and  also  through  the  points  A  and  B. 

3.  Take  any  four   points,  A,  B, 

C,  D,  on  the  circumference  of  a 
circle,  and  join  them  as  in  the 
figure,  so  constructing  a  quadri- 
lateral in  the  circle.  Adjust  the 
bevel  to  the  opposite  angles  B  and 

D,  and  construct  angles  equal  to 
them  adjacent  to  one  another. 
What  do  you  observe  with  refer- 
ence to  the  sum  of  the  angles  B 
and  D?  What  with  reference  to 
the  sum  of  the  angles  A  and  C  ? 

Repeat   this  measurement   with 
respect  to  the  opposite   angles  of  other   quadrilaterals 
in  circles. 


88 


Geometry. 


The  annexed  figure  suggests 
wliat  conclusion  should  be  reached 
with  respect  to  the  sum  of  the 
angles  at  B  and  D  and  at  A  and 
C.  For  the  angle  marked  at  0  is 
double  of  the  angle  ADC  (Ch.  XII., 
1)  J  and  the  other  angle  at  0  is 
double  the  angle  ABC.  Therefore 
the  angles  at  0  are  together  double 

the  sum  of  the  angles  ADC  and  ABC.  But  the  angles 
at  0  make  up  four  right  angles.  Hence  the  angles 
ABC,  ADC  are  together  equal  to  two  right  angles. 

Hence  the  opposite  angles  of  a  quadrilateral 
inscribed  in  a  circle  are  together  equal  to  two 
right  angles. 

Using  the  protractor,  con- 
struct a  quadrilateral  with  two 
of  its  opposite  angles  together 
equal  to  two  right  angles. 
Taking  any  three  of  the  angu- 
lar points,  and,  by  the  method 
of  Ch.  X.,  6,  constructing  for  the  circle  through  these 
three  points,  and  describing  the  circle,  note  the  position 
of  the  quadrilateral  with  respect  to  the  circle. 

Repeat  the  construction  for  several  such  quadrilat- 
erals. 

The  result  of  such  observations  is  that  if  the  op- 
posite angles  of  a  quadrilateral  are  together 
equal  to  two  right  angles,  a  circle  can  be 
described  about  it. 


Angles  in  a  Circle. 


89 


Since  a  quadrilateral  can  be  divided  into  two  tri- 
angles by  joining  its  opposite  angles,  the  sum  of  all 
the  angles  of  any  quadrilateral  is  four  right  angles. 
Hence  if  the  sum  of  a  pair  of  opposite  angles  be  two 
right  angles,  the  sum  of  the  other  pair  is  two  right 
angles  also. 

4.  Describe  a  circle,  and  in 
the  semicircle  construct  a  num- 
ber of  angles  as  indicated  in  the 
figure.  Adjust  the  protractor  to 
the  angles  ADB,  AEB,  .... 
What  is  the  magnitude  of  these 
angles  ? 

The  magnitude  of  the  angle 
in  a  semicircle  may  be  proved  thus  :  The  straight 
angle  ACB  at  the  centre  is  (Ch.  XII.,  1)  double  any  of 
the  angles  at  the  circumference.  But  the  straight  angle 
ACB  is  180°.  Hence  the  angle  in  a  semicircle  is 
90°. 

ADB  being  a  right-angled 
triangle,  find  the  centre  of 
the  circle  through  A,  D  and 
B,  by  bisecting  AD,  BD  and 
drawing  the  perpendiculars 
EC,    FC.     Note   that   these 

perpendiculars  intersect  in  AB ;  and  note  also  that  C, 
being  the  centre  of  the  circle  through  A,  D  and  B,  the 
centre  of  the  hypotenuse  of  a  right-angled  triangle  is 
equidistant  from  the  three  angles  of  the  triangle. 


90 


GrEOMETKY. 


5.  A  chord,  such  as  AB,  which 
does  not  pass  through  the  centre, 
divides  the  circle  into  two  seg- 
ments, one  of  which,  ADB,  is 
greater,  and  the  other,  ACB,  less 
than  a  semicircle.  Evidently  the 
marked  angle  AOB  is  greater 
than  two  right  angles,  and  there- 
fore   the    angle    ACB,   which   is 

half  of  the  marked  angle  AOB,  is  greater  than  one 
right  angle.  Similarly  the  angle  ADB,  being  half  the 
other  angle  at  0,  is  less  than  a  right  angle. 

Hence  the  angle  in  a  segment  of  a  circle 
less  than  a  semicircle  is  greater  than  a  right 
angle ;  and  the  angle  in  a  segment  of  a  circle 
greater  than  a  semicircle  is  less  than  a  right 
angle. 

AC,  CB  contain  an  angle  which  has,  in  succession, 
the  magnitudes  80°,  85°,  89°,  91°,  95°.  Construct  in 
the  different  cases  the  circles  through  A,  C  and  B, 
and  note  the  positions  of  the  centre  with  respect  to 
the  side  AB. 

6.  Draw  with  accuracy  the  tan- 
gent CAB  at  any  point  A  on  the 
circumference  of  a  circle.  From 
A  draw  any  chord  AD,  and  con- 
struct the  angles  AED,  AFD  in 
the  segments  into  which  AD 
divides  the  circle.  Then,  using 
the  bevel,  discover  the  relation  in 
size  between  the  angle  CAD  and  the  angle  AFD  in  the 
alternate  segment;  and  the  relation  between  the  angle 
BAD  and  the  angle  AED  in  the  alternate  segment. 


Angles  in  a  Cikcle.  91 

Repeat  the  same  examination  in  the  case  of  different 
circles,  drawing  the  chord  at  various  inclinations  to 
the  tangent. 

As  a  result  of  these  observations  we  are  led  to  the 
conclusion  that  if  from  the  point  of  contact  of 
any  tangent  to  a  circle,  a  chord  be  drawn  cut- 
ting the  circle,  the  angles  the  chord  makes 
with  the  tangent  are  equal  to  the  angles  in 
the  alternate  segments  of  the  circle. 

We  may  establish  the  same  re- 
sult in  the  following  way:  Let 
AG  be  the  diameter  through  A. 
The  angles  GED,  GAD,  GFD,  are 
equal  to  one  another  because  they 
stand  on  the  same  arc  GD.  Also 
the  angles  CAG,  BAG,  AEG,  AFG 
are  right  angles.     Hence 

ZBAG-ZDAG=ZAEG-  ZDEG, 

or  ZBAD=ZAED,  in  alternate  segment. 
Again,    L  CAG  +  L  DAG  =  L  AFG  +  Z  GFD, 

or  L  CAD  =  L  AFD,  in  alternate  segment. 

It  will  be  noticed  that,  as  AD  revolves  to  the 
right  about  A,  the  angles  BAD,  AED,  have  just  as 
much  taken  from  them  as  CAD,  AFD  have  added  to 
them,  the  points  E  and  F  being  supposed  to  remain 
stationary. 

Placing  the  centre  of  the  protractor  on  the  circum- 
ference of  a  circle,  and  marking  the  initial  line  of 
protractor  as  a  chord,  we  may  place  in  the  circle  an 
angle  of  any  required  magnitude,  i.e.,  we  may  cut  off 
from  the  circle  a  segment  containing  an  angle  of  any 
size. 


92  Geometky. 


Exercises. 

1.  Describe  a  circle  of  radius  IJ  in.,  and  in  it  place  an  angle  of  60°. 
In  it  also  describe  a  triangle  of  vertical  angle  60°  and  altitude  2  in. 

2.  Describe  a  circle  of  radius  35  millimetres.  From  it  cut  off  a 
segment  containing  an  angle  of  50°,  and  describe  in  it  a  triangle  with 
angles  50°,  30°  and  100°. 

3.  Describe  a  circle  of  radius  40  millimetres,  and  in  it  describe  a 
triangle  with  angles  50°,  55°  and  75°. 

4.  Describe  a  circle  of  radius  2  in.  Draw  a  chord  AB,  cutting  off 
a  segment  containing  an  angle  of  120°,  and  a  chord  BC,  cutting  off  a 
segment  containing  an  angle  of  100°.  What  is  the  angle  contained 
in  the  segment  cut  off  by  CA  ?    Apply  test.     Give  reason, 

5.  Describe  a  circle  of  radius  50  millimetres,  and  in  it  draw  a  chord 
cutting  off  a  segment  containing  an  angle  of  55°.  What  angle  is 
contained  in  the  segment  which  forms  the  rest  of  the  circle  ?  Apply 
test.     Give  reason. 

6.  Describe  a  circle  of  radius  If  in.  Draw  in  it  a  chord  AB, 
dividing  the  circle  into  two  segments,  ACB,  ADB,  containing  angles 
of  70°  and  110°  respectively.  Construct  in  the  circle  an  angle  CAD 
of  50°.  What  is  the  angle  CBD  ?  Mark  on  the  quadrilateral  ACBD 
the  size  of  each  angle. 

7.  Describe  a  circle  of  radius  40  millimetres,  and  in  it  construct  a 
quadrilateral  with  angles  55°,  75°,  125°,  105°. 

8.  Describe  a  circle  of  radius  45  millimetres,  and  in  it  draw  a 
number  of  chords,  AB,  CD,  EF,  ...  all  cutting  off  angles  of  60°. 
Are  the  chords  all  of  the  same  length  ?     Apply  test.     Give  reasons. 

9.  Describe  a  circle  of  radius  1^  in.,  and  in  it  construct  a 
triangle  with  angles  30°,  70°,  80°.  Does  the  size  of  the  triangle  vary 
according  as  it  happens  to  be  placed  in  the  circle  ?     Give  reasons. 

10.  Describe  a  circle  of  radius  2  in.,  and  in  it  construct  a  quadri- 
lateral with  angles  45°,  120°,  135°,  60°.  Show  that  the  size  and  shape 
of  the  quadrilateral  can  be  made  to  vary.  What  lines  belonging  to 
the  quadrilateral  remain  constant  ? 

11.  A  BCD  is  a  quadrilateral  in  a  circle,  and  the  side  AB  is  pro- 
duced to  E.  To  what  angle  of  the  quadrilateral  is  the  exterior  angle 
CBE  equal  ?    Apply  test.     Give  reasons. 


Exercises.  93 

•  12.  AB  is  a  line  of  length  2J  in.  If  on  it  a  segment  of  a  circle  is  to 
be  constructed  containing  an  angle  of  60°,  what  angle  will  AB  sub- 
tend at  the  centre  C  ?  What  are  the  angles  of  the  triangle  CAB  ? 
Find  C  by  construction,  and  then  describe  the  circle. 

13.  AB  is  a  line  of  length  60  millimetres.  Following  the  method 
suggested  in  the  previous  question,  construct  on  it  a  segment  of  a 
circle  containing  an  angle  of  70°.  Test  the  accuracy  of  your  con- 
struction by  measuring  an  angle  in  the  segment. 

14.  AB  is  a  line  of  length  2^  in.  ;  to  construct  on  it  a  segment  of  a 
circle  containing  an  angle  of  70° :  Make  BAC  =  90°,  ABC  =  90°  -  70°  = 
30°.  Then  ACB  =  70°.  Bisect  BC  at  0,  and  with  O  as  centre  and 
OA,  OB,  or  OC  as  radius,  describe  a  circle.  The  segment  ACB  con- 
tains an  angle  of  70°,  and  it  stands  on  AB. 

15.  Construct  a  triangle  with  sides  60,  75  and  85  millimetres.  On 
these  sides,  and  within  the  triangle,  construct  segments  containing 
angles  of  120°.  Should  these  segments  all  pass  through  the  same 
point  within  the  triangle  ? 

16.  AB,  CD  are  two  chords,  perpendicular  to  each  other,  in  a  circle 
whose  centre  is  O.  Of  what  angles  are  the  angles  AOC,  BOD 
double  ?     What,  therefore,  is  their  sum  ? 

17.  AB,  CD  are  two  chords  of  a  circle,  intersecting  in  E.  Show 
that  the  triangles  AEC,  DEB  are  equiangular. 

18.  ABCD  is  a  quadrilateral  in  a  circle,  and  the  sides  AB,  CD, 
produced,  meet  in  E.  Show  that  the  triangles  EBC,  EDA  are  equi- 
angular. 

19.  AB,  AC  are  tangents  to  a  circle  whose  centre  is  O.  Show 
that  BOC  =  180  — A  ;  also  that  the  angle  in  the  segment  BC,  between 
the  tangents,  contains  an  angle  90°  +  a  A- 

20.  AD,  BE  are  drawn  perpendicular  to  the  opposite  sides  of  the 
triangle  ABC.  Show  that  a  circle  can  be  described  about  AEDB, 
and  describe  it.  How  are  the  angles  ABC,  DEC  related  ?  Apply 
test.     Give  reasons. 


CHAPTER  XIII. 


B 


Relation  Between  Segrments  of  Intersecting" 
Chords. 

1.  AEB  and  CED  are  any  two 
chords  in  a  circle,  intersecting 
at  E. 

In  the  second  figure  CEB  and 
AED  are  any  two  lines  drawn 
perpendicular  to  each  other,  and 
and  on  these  we  lay  off  the 
following  distances  with  the 
dividers : 

AE  =  AE  of  circle 

EB  =  EB    ''      '' 

CE=CE    ^'      '' 

ED  =  ED   '^      '' 
Complete  the   rectangles   CEDF 
and  AEBG,  and  let  FD  and  GB 

meet  in  H.  Then  produce  the  lines  FC^  HE  and  GA, 
and  note  how  nearly  they  come  to  passing  through 
the  same  point  (at  K).  Go  over  the  measurements  and 
construction  with  extreme  care,  getting  rid  of  all  inac- 
curacies. Do  these  lines  (FC,  HE,  GA)  all  pass  through 
the  same  point?  If  they  do,  how  do  the  areas  CEDF, 
AEBG  compare  in  size  (Ch.  YIII.,  6),  and  therefore  the 
rectangles  AE.EB,  CE.ED,  contained  by  the  segments 
of  the  chords  ? 

Measure  the  number  of  millimetres  in  each  of  the 
lines  AE,  EB,  CE,  ED  in  the  circle,  and  examine 
whether  the  product  of  AE  and  EB  is  approximately 
equal  to  the  product  of  CE  and  ED. 

94 


E 


Segments  of  Chokds  of  Cikcles. 


95 


Describe  other  circle  s,  draw  two  chords  in  each,  and 
repeat  in  the  case  of  each  circle  the  construction  of 
the  second  figure.  Repeat  also  the  measurements  and 
multiplications. 

The  result  of  our  observations  may  be  stated  as 
follows :  If  two  chords  of  a  circle  cut  one  an- 
other within  the  circle,  the  rectangle  contained 
by  the  segments  of  the  one  is  equal  to  the 
rectangle  contained  by  the  segments  of  the 
other. 

2.  Draw  accurately  the  tangent  EC ; 
draw  also  the  secant  EAB. 

In  the  second  figure  CEA,  BEC  are 
any  two  lines  drawn  at  right  angles  to 
each  other,  and  on  these  we  lay  off  the 
following  distances  with  the  dividers: 

EA  =  EA  of  circle 
EB  =  EB   "       '' 
EC,  EC  =  EC   ^'       " 

Complete  the  rectangle  EBGA  and  the 
square  ECFC,  and  let  FC,  GA  meet 
in  H.  Then  produce  the  lines  FC, 
HE  and  GB,  and  note  how  nearly  they 
come  to  passing  through  the  same 
point  (at  K).  Go  over  the  measure- 
ments and  construction  with  extreme 
care,  getting  rid  of  all  inaccuracies. 
Do  these  lines  (FC,  HE,  GB)  all  pass  through  the  same 
point?  If  they  do,  how  do  the  areas  EBGA,  ECFC 
compare  in  size  (Ch.  VIII.,  6),  and  therefore  the  rect- 
angle EA.EB  and  square  on  EC  (see  figure  of  circle)  ? 


96  Geometry. 

Measure  the  numbers  of  millimetres  in  each  of  the 
lines  EA,  EB,  EC  in  the  first  figure,  and  examine 
whether  the  product  of  EA  and  EB  is  approximately- 
equal  to  the  square  of  EC. 

Describe  other  circles,  draw  to  each  a  secant  and  a 
tangent  from  the  same  point,  and  repeat  in  the  case 
of  each  the  construction  of  the  second  figure.  Repeat 
also  the  measurements  and  multiplications. 

The  result  of  our  observations  may  be  stated  as 
follows :  If  from  any  point  without  a  circle  two 
straight  lines  be  drawn,  one  a  secant  and  the 
other  a  tangent,  then  the  rectangle  contained 
bj'-  the  secant  and  the  part  of  it  without  the 
circle  is  equal  to  the  square  on  the  tangent. 

If  another  secant  EDF  be 
drawn,  since  the  rectangle  con- 
tained by  EA  and  EB  is  equal  to 
the  square  on  EC,  and  the 
rectangle  contained  by  ED  and 
EF  is  equal  to  the  square  on 
EC,  therefore  the  rectangle  con- 
tained by  EA  and  EB  is  equal  to 
the  rectangle  contained  by  ED 
and  EF. 

The  segments  of  one  chord  are 
3,  4,  and  of  another  2,  6  quarters 

of  an  inch,  the  chords  making  an  angle  of  30°  with 
one  another,  describe  the  circle  through  the  ex- 
tremities of  the  chords.  If  the  segments  of  another 
line  through  the  intersection  of  the  chords  be  IJ  and 
8  quarters  of  an  inch,  do  the  ends  of  this  necessarily 


Exercises.  97 

rest  on  the  circle?     Place  the  line  that  its   ends  may 
so  rest. 

The  tangent  to  a  circle  is  60  millimetres  ;  a  secant 
is  90^  and  the  part  of  it  without  the  circle  40 
millimetres.  These  lines  make  an  angle  of  60°  with 
one  another.     Describe  the  circle. 


Exercises. 

1.  Two  lines  AB,  CD  intersect  in  E.  AE  =  30,  EB  =  40,  CE  =  20, 
ED=60  millimetres,  so  that  AE.EB  =  CE.ED.  Show  that  a  circle 
can  be  described  to  pass  through  the  four  points  A,  C,  B,  D,  i.e.,  that 
a  circle  through  A,  D,  B,  say,  also  passes  through  C. 

2.  Two  lines,  AB,  CD  cut  one  another  in  E.  AE  =  1|,  EB  =  2, 
CE  =  3,  ED  =  1  in.,  so  that  AE.EB  =  CE.ED.  Describe  a  circle  to 
pass  through  A,  C,  B,  D. 

3.  Describe  a  circle  of  radius  2  in.  Draw  a  diameter  AB.  Take 
in  it  a  point  G  at  distance  1  in.  from  centre,  and  draw  chord  DCE 
perpendicular  to  AB.  By  construction,  as  in  text,  show  that 
rectangle  AC.CB  is  equal  to  square  on  CD. 

It  may  also  be  shown  that  CD=  ^^3  in.  by  proving  it  equal  to 
the  altitude  of  an  equilateral  triangle  whose  side  is  2  in. 

4.  Describe  a  circle  of  radius  in  2\  in.  Draw  a  diameter  AB.  In 
it  take  a  point  C  at  distance  1|  in.  from  centre,  and  draw  chord 
DCE  perpendicular  to  AB.  What  should  be  the  length  of  CD? 
Measure  it. 

5.  Two  lines  intersect  at  an  angle  of  30°.  The  segments  of  one 
2  in.  and  \  in.,  of  the  other,  both  1  in.  Describe  a  circle  to  pass 
through  the  ends  of  the  lines.  With  what  inclination  of  the  lines  to 
one  another  would  the  longer  line  become  a  diameter  ? 

6.  Describe  a  circle  of  radius  3  in.  In  it  place  a  chord  of  length 
4  in. ,  and  take  in  the  chord  a  point  at  distance  1  in.  from  an  end. 
Through  this  point  draw  another  chord  whose  segments  shall  be  l^in. 
and  2  in. 

7.  Describe  a  circle  of  radius  70  millimetres.  In  it  place  a  chord 
of  length  90  millimetres,  and  take  a  point  in  the  chord  at  distance 


98  Geometky. 

40  millimetres  from  an  end.     Through  this  point  draw  two  chords 
whose  segments  shall  be  20  and  100  millimetres. 

8.  On  a  line  take  lengths,  AB,  AC,  of  27  and  48  millimetres,  in 
the  same  direction.  Draw  a  line  AD  of  36  millimetres,  making  an 
angle  of  45°  with  AC.  Describe  a  circle  through  B,  C,  D.  What  is 
AD  with  respect  to  this  circle  ? 

9.  Same  problem  as  previous,  but  with  AB  =  36,  AC  =  64,  AD  =  48 
millimetres,  and  angle  between  AC,  AD,  60°.  Describe  a  circle 
through  B,  C  and  D.  What  position  does  AD  occupy  with  respect 
to  it? 

10.  AB,  AC,  measured  along  the  same  line,  in  the  same  direction, 
are  36  and  64  millimetres  ;  and  AD  another  line  through  A  is  48 
millimetres.  Place  AD  so  that  the  circle  through  B,  C  and  D  may- 
have  its  centre  in  AC. 

11.  AB,  AC  measured  along  the  same  line,  in  the  same  direction, 
are  18  and  72  millimetres.  Describe  a  number  of  circles  through  B 
and  C,  and  from  A  draw  a  tangent  to  each.  Measure  the  lengths  of 
these  tangents.     What  relation  between  the  lengths  and  why  ? 

12.  Two  lines  AB,  AC  of  length  ^Z  in. ,  both  touch  the  same  circle 
at  B  and  C,  and  make  an  angle  of  60°  with  one  another.  Construct 
the  circle.     What  is  its  radius  ? 

13.  AB,  AC  measured  along  the  same  line  in  the  same  direction  are 
48  and  108  millimetres.  Describe  a  circle  on  BC  as  diameter,  and 
draw  a  line  ADE  cutting  the  circle  in  D  and  E,  such  that  AD  =  54 
millimetres.  What  is  the  length  of  AE  ?  Draw  a  tangent  to  the 
circle  from  A.     What  is  its  length  ? 

14.  Describe  two  circles  of  radii  1  and  2  inches  respectively,  inter- 
secting in  A  and  B.  Draw  a  straight  line  through  A  and  B,  and 
from  any  point  in  it,  draw  a  tangent  to  each  circle.  Measure  the 
tangents.     What  relation  between  their  lengths  ?     Give  reason. 

15.  Describe  two  circles  of  radii  25  and  70  millimetres,  intersecting 
in  A  and  B.  Draw  a  straight  line  through  A  and  B,  and  from  any 
point  in  it  draw  a  tangent  to  each  circle.  Measure  the  tangents. 
What  relation  between  their  lengths  ?     Give  reason. 

16.  Describe  three  circles  of  radii  2,  3  and  3|  inches,  so  that  each 
intersects  the  other  two.  Through  each  pair  of  points  of  intersection 
draw  straight  lines.  These  three  lines  should  pass  through  the  same 
point. 


EXEKCISES.  99 

17.  If  the  tangents  to  two  intersecting  circles  from  any  point  be 
equal,  that  point  must  lie  on  the  line  joining  the  points  of  intersec- 
tion of  the  circles. 

18.  The  common  chord  of  two  intersecting  circles  on  being  pro- 
duced, cuts  a  line  that  touches  both  circles.  Show  that  the  tangent 
line  must  be  bisected. 

19.  ABC  is  a  triangle  right-angled  at  C,  and  from  C  a  perpendicular 
CD  is  drawn  to  AB.  By  describing  a  circle  about  ABC,  show  that 
the  rectangle  AD.  DB  is  equal  to  the  square  on  CD. 

20.  ABC  is  a  triangle  right-angled  at  C,  and  from  C  a  perpen- 
dicular CD  is  drawn  to  AB.  By  describing  a  circle  about  the 
triangle  CDB,  show  that  the  rectangle  AD.AB  is  equal  to  the  square 
on  AC. 

21.  In  the  previous  question,  describe  a  circle  about  the  triangle 
ACD,  and  show  that  the  rectangle  BA.  BD  is  equal  to  the  square  on 
BC. 

22.  The  sides  of  a  triangle  are  3,  4,  5,  and  a  perpendicular  is 
dropped  from  the  right  angle  in  the  hypotenuse.  Find  the  lengths 
of  the  segments  of  the  hypotenuse  on  each  side  of  the  perpendicular, 
and  also  the  length  of  the  perpendicular. 


CHAPTl^R  XIV. 

Triangles  In  and  About  Circles. 

1.  A  triangle  is  said  to  be  inscribed  in  a 
circle  when  the  three  angular  points  of  the 
triangle  rest  on  the  circumference  of  the  circle. 

We  evidently  cannot  in  general  construct  in  a  circle 
of  given  size  a  triangle  equal  to  a  given  triangle.  In 
a  small  circle  we  could  not  place  a  large  triangle. 
Indeed  we  have  seen  (Ch.  X.,  6)  that  there  is  but  one 
circle  which  can  be  made  to  fit  round  a  triangle  of 
given  size. 

We  can,  however,  always  inscribe  in  any 
circle  a  triangle  equiangular  to  another  tri- 
angle, i.e.,  a  triangle  with  its  angles  of  given  size, 
their  sum  of  course  being  180°.  Thus  let  it  be  re- 
quired to  construct  in  a  given  circle  a  triangle  whose 
angles  shall  be   30°,  70°,  80°. 

Using  the  protractor,  adjust  the  bevel  to  an  angle 
equal  to  any  one  of  these,  say,  30°.  Place  the  angle 
of  the  bevel  at  any  point  C  on  the  circumference,  and 
with  a  needle  mark  the  points, 
A  and  B,  where  the  legs  of  the 
bevel  cross  the  circumference. 
We  have  thus  a  segment  ACB 
containing  an  angle  of  30°,  and 
all  angles  in  the  segment  ACB 
are  angles  of  30°.  With  the 
protractor  at  A  make  the  angle 
BAD  of  80°.  Join  BD.  Then 
ADB  is  an  angle  of  30°.  Hence  the  remaining  angle 
ABD  is  70°. 

Of  course  the  angle  of  30°  at  C  may  be  constructed 

100 


Triangles  In  and  About  Circles.        101 

with  the  protractor.  The  segment  QoMmniiig  an  angle 
of  30°  may  also  be  obtained  hy  constynstiag;  a^^  the; 
centre  an  angle  of  60°.  -   .''.::,.  :':      -'    '   -    '  "' 

In  a  circle  whose  radins  is  45  millimetres,  construct 
a  triangle  whose  angles  are  75°,  45°  and  60°. 

In  a  circle  whose  radius  is  IJ  in.,  construct  a  tri- 
angle whose  angles  are  65°,  75°  and  40°. 

2.  To  construct  a  triangle  whose  sides  shall 
be  tangents  to  a  given  circle,  and  whose 
angles  shall  be  of  given  magnitude,  say,  75°, 
45°  and  60°. 

We  can  scarcely  here  proceed  as  in  the  previous 
case,  adjusting  the  legs  of  the  bevel  to,  say,  the  angle 
75°,  and  placing  them  across  the  circle  so  as  to  be 
tangents  to  it.  To  assume  that  we  can  construct  the 
tangent  to  a  circle  by  laying  the  ruler  against  it  and 
so  drawing  a  line,  is  equivalent  to  assuming  that  we 
can  lay  off  a  right  angle,  using  only  the  judgment  of 
the  eye. 

It  will  be  well  to  proceed  thus:  Find  the  angles 
which  are  the  supplements  of  75°,  45°  and  60°,  i.e., 
105°,  135°  and  120°.  Draw 
any  radius  OA,  and  make  the 
angle  AOB  of  105°,  and  the 
angle  AOC  of  135°.  The  re- 
maining angle  BOC  must  be  of 
120°,  since  105°  + 135°  +  120° 
=  360°.  Draw  lines  (tangents) 
at  A,  B  and  C  at  right  angles 
to  the  radii. 

Since  the  angles  of  a  quadrilateral  make  up  four 
right  angles,  and  the  angles  at  A  and  C  are  right 
angles,   therefore   AOC  + AEC  =  180°.     But  AOC  is   135°. 


102 


Geometry. 


Therefore  A^C  is '  ^5°,  if  AOC  has  been  accurately 
C(>i>strueted,«  and .  tlie '  tangents  at  A  and  C  correctly 
drawn,  i:^imilarly  the  angles  at  D  and  F  are  60°  and 
75°  respectively. 

The  triangle  DEF  is  said  to  have  been  de- 
scribed about  the  circle. 

About  a  circle  whose  radius  is  20  millimetres,  con- 
struct a  triangle  whose  angles  are  70°,  80°  and  30°. 

About  a  circle  whose  radius  is  35  millimetres,  con- 
struct a  triangle  whose  angles  are  90°,  30°,  and  60°. 

About  a  circle  whose  radius  is  IJ  in.,  construct  an 
equilateral  triangle. 

About  a  circle  whose  radius  is  IJ  in.,  construct  an 
isosceles  triangle  whose  vertical  angle  is  30°. 

3.  In  a  circle  we  readily  place 
a  chord  of  any  required  length. 
For,  take  the  length  on  the 
ruler  with  the  points  of  the 
dividers,  and  place  the  points  of 
the  dividers  on  the  circumfer- 
ence of  the  circle.  The  ends 
of  the  chord,  A,  B  are  thus 
marked,  and  the  chord  can  be  drawn. 

We  can  without  difficulty 
draw  the  chord  in  a  re- 
quired position,  for  exam- 
ple, parallel  to  a  given 
line,  KL :  Draw  OC  per- 
pendicular to  KL,  and 
mark  off  CD,  CE  each 
equal  to  half  the  length 
of  the  chord.  Then  draw 
DB,    EA,    parallel    to    CO. 


Exercises.  103 

The  chord  AB  is  equal  to  ED,  and  therefore  is  of  the 
required  length,  and  it  is  parallel  to  KL. 

We  may  draw  EA  alone  perpendicular  to  KL,  and 
then  draw  AB  parallel  to  KL,  thus  not  using  the  point 
D  or  line  DB. 

Of  course  the  chord  can  never  be  greater  than  the 
diameter  of  the  circle  in  which  it  is  to  be  placed. 

In  a  circle  whose  radius  is  55  millimetres,  draw 
chords,  with  one  end  at  the  same  point,  of  lengths  20, 
25,  30,  35,  40,  45,  50,  55  and  110  millimetres. 

In  a  circle  of  radius  1  inch,  place  ten  chords  of 
length  J  inch,  such  that  each  ends  at  the  point  where 
the  next  begins. 

In  a  circle  of  radius  -30  millimetres,  place  six  chords 
each  of  length  30  millimetres,  such  that  each  ends 
where  the  next  begins. 

In  a  circle  place  a  chord  of  given  length  so  that  it 
may  be  perpendicular  to  a  given  line. 

Exercises. 

1.  In  a  circle  of  radius  45  millimetres,  place  an  angle  of  35°;  also 
an  angle  of  145°. 

2.  In  the  circle  of  the  previous  question  place  these  same  angles  so 
that  the  chord  or  chords  on  which  they  stand  may  be  parallel  to  a 
line  that  makes  45°  with  the  edge  of  your  paper. 

3.  In  a  circle  of  radius  2  in.,  place  an  angle  of  50°,  so  that  the 
chord  on  which  it  stands  may  be  perpendicular  to  a  line  that  makes 
an  angle  of  60°  with  the  edge  of  your  paper. 

4.  In  a  circle  of  radius  1  in.,  place  in  succession  four  chords,  AB, 
BC,  .   .   .  ,  each  of  length  J2  in. 

5.  In  a  circle  of  radius  1^  in,,  construct  an  equilateral  triangle. 

6.  In  a  circle  of  radius  2  in.,  construct  an  isosceles  triangle,  the 
angle  at  the  vertex  being  55°.  (Construct  at  centre  an  angle  of  110°. 
The  symmetry  of  the  circle  suggests  the  rest  of  the  construction. ) 


104  Geometry. 

7.  In  a  circle  of  diameter  3^  in.,  construct  an  equilateral  triangle, 
such  that  its  base  shall  be  parallel  to  the  top  or  bottom  of  your  paper. 
(Draw  a  line  through  centre  perpendicular  to  top  or  bottom  of  paper, 
and  at  centre  construct,  on  each  side  of  this  line,  angles  of  60°.    Etc. ) 

8.  Construct  a  triangle  with  angles  of  55°,  65°,  and  60°,  and  in  a 
circle  whose  radius  is  1|  in.  construct  a  triangle  equiangular  to  this, 
its  sides  being  also  parallel  to  the  sides  of  this  triangle. 

9.  Describe  a  circle  of  radius  48  millimetres,  and  draw  a  line 
making  an  angle  of  45°  with  the  edge  of  your  paper.  Construct  a 
triangle  with  angles  48°,  75°,  and  57°,  so  that  the  side  opposite  48° 
may  be  parallel  to  the  line. 

10.  Describe  a  circle  of  radius  40  millimetres,  and  draw  a  line 
making  an  angle  of  60°  with  the  side  of  your  paper.  Draw  a  tangent 
to  the  circle  parallel  to  this  line.  (From  centre  drop  a  perpendicular 
on  the  line.     This  gives  point  through  which  tangent  is  to  be  drawn.  ) 

11.  Describe  a  circle  of  radius  35  millimetres.  Draw  a  line  making 
an  angle  of  75°  with  the  top  or  bottom  of  your  paper,  and  draw  a 
tangent  to  the  circle  perpendicular  to  this  line.  (Draw  perpendicular 
to  line,  and  then  tangent  parallel  to  this  perpendicular. ) 

12.  About  a  circle  of  radius  1  in.  describe  an  equilateral  triangle. 

13.  Describe  a  circle  of  radius  35  millimetres,  and  about  it  describe 
an  equilateral  triangle  so  that  two  of  the  sides  may  make  angles  of 
60°  with  the  side  of  your  paper,  the  third  side  being  parallel. 

14.  Describe  a  circle  of  radius  25  millimetres,  and  about  it  describe 
an  isosceles  triangle  whose  vertical  angle  is  40°,  the  base  of  the 
triangle  being  parallel  to  the  top  or  bottom  of  your  paper. 

15.  About  a  circle  of  radius  IJ  in.  describe  a  triangle  whose  angles 
are  30°,  70°  and  80°. 

16.  Draw  any  three  intersecting  lines.  Describe  a  circle  of  radius 
li^e"  in.,  and  about  it  describe  a  triangle  whose  sides  are  parallel  to 
the  lines.  Test  the  accuracy  of  your  construction  by  comparing  the 
angles  of  the  two  triangles. 

17.  When  a  triangle  ABC  is  inscribed  in  a  circle,  what  are  the 
magnitudes  of  the  angles  which  the  sides  subtend  at  the  centre  com- 
pared with  the  magnitudes  of  the  angles  of  the  triangle  ? 


Exercises.  105 

18.  Describe  a  circle  of  radius  1^  in.  In  and  about  it  describe  two 
triangles  with  angles  50°,  60°  and  70°,  so  that  corresponding  sides  are 
parallel  to  each  other. 

19.  An  equilateral  triangle  is  inscribed  in  a  circle,  and  another  is 
described  about  the  circle.  What  relation  exists  between  the  lengths 
of  the  sides  ? 

20.  Describe  a  circle  of  radius  32  millimetres,  and  draw  two 
tangents  to  it,  such  that  the  angle  between  them  is  25°. 

21.  Describe  a  circle  of  radius  1  in,,  and  from  the  same  point 
draw  two  tangents  to  the  circle,  each  of  length  3  in. 

22.  Describe  two  circles  of  radii  1  in.  and  2  in.  In  them  describe 
triangles  with  angles  of  45°,  65°  and  70°.  Compare  the  lengths  of 
corresponding  sides  of  the  two  triangles. 


CHAPTER  XV. 

Circles  In  and  About  Triangrles. 

1.  If  the  angle  BAG,  between  two  lines,  be  bisected, 
and,  from  any  point 

D  in  it,  perpendicu- 
lars DB,  DC  be  drawn, 
these  perpendiculars 
are  evidently  equal. 
If,  then,  a  circle  be 
described  with  centre 
D,  and  radius  DB  or  DC,  it  wiU  touch  both  the  lines. 

Thus  all  circles  touching  both  lines  have 
their  centres  in  the  straight  line  which  bisects 
the  angle  between  the  lines. 

Two  lines  make  an  angle  of  120°  with  one  another. 
Describe  four  circles,  of  different  radii,  touching  both 
of  them. 

Two  lines  make  an  angle  of  80°  with  one  another. 
Describe  a  circle  of  radius  ^  in.  to  touch  both  of  them  j 
also  of  radius  1  in. 

Two  lines  make  an  angle  of  60°  with  one  another. 
Describe  a  circle  touching  both  of  them ;  also  a  second 
circle  touching  the  previous  circle  and  the  two  lines. 

2.  We  may  describe  a  circle  touching  the 
three  sides  of  a  triangle  as  follows : 

106 


CiKCLES  In  and  About  Triangles.        107 


Hence  with  centre  D 
This  is  the    circle 


Bisect  the  angles  at  B  and  C 
by  the  lines  BD,  CD.  Then 
BD  contains  the  centres  of 
circles  touching  BA  and  BC; 
and  CD  contains  the  centres 
of  circles  touching  CA  and  CB. 
Hence  D  is  the  centre  of  a 
circle  which  touches  all  three 
sides.  DE,  perpendicular  to 
BC,  is  the  radius  of  this  circle. 
and  radius  DE,  describe  a  circle. 
inscribed  in  the  triangle  ABC. 

The  utmost  care  is  to  be  exercised  in  accurately  bisect- 
ing the  angles;  otherwise  it  may  be  found  that,  when 
the  circle  is  described,  it  cuts  a  side,  or  falls  short 
of  one. 

Inscribe  a  circle  in  the  triangle  whose  sides  are  75, 
80  and  95  millimetres. 

Describe  a  circle  to  touch 
*(any  triangle  ABC),  and  the 
duced. 

The  base  of  a  triangle   is  2  in.,    and    the   angles 
the   base    are   40°    and   110°. 
Measure  its  radius. 


the    other    side    of    BC 
sides  AB    and    AC    pro- 


Inscribe  a  circle   in 


at 
it. 


3.  We  have  already  (Ch.  X.,  6), 
in  effect,  shown  how  to  describe 
a  circle  about  any  triangle, 

i.e.,  to  pass  through  the  angular 
points  of  the  triangle.  Two 
sides,  say  AB  and  AC,  are  bi- 
sected, and  DO,  EO  are  drawn 
through   the  points  of  bisection 


108  Geometry. 

perpendicular  to  AB  and  AC,  respectively.  Then  all 
points  in  DO  are  equally  distant  from  A  and  B ;  and 
all  points  in  EO  are  equally  distant  from  A  and  C. 
Hence  0  is  equally  distant  from  A,  B  and  C  ;  and  if 
the  sharp  point  of  the  compasses  be  placed  at  0,  and 
the  pencil  end  at  A,  or  B,  or  C,  and  a  circle  be  de- 
scribed,  it  will   pass  through   A,  B  and  C. 

Here  again  the  greatest  care  must  be  exercised  in 
bisecting  the  sides,  and  in  drawing  the  perpendiculars 
at  the  points  of  bisection ;  otherwise  the  circle  will 
pass  through  the  angle  on  which  the  pencil  end  of  the 
compasses  was  placed,  but  may  not  pass  through  the 
two  other  angles. 

Describe  a  circle  about  a  triangle  whose  sides  are 
55,  70  and  90  millimetres.     Measure  its  radius. 

The  side  of  an  equilateral  triangle  is  3  in.  j  describe 
a  circle  about  it. 

Each  of  the  equal  sides  of  an  isosceles  triangle  is  3 
in.,  and  the  equal  angles  are  each  75°.  Describe  a 
circle  about  it. 


Should  the  course  contained  In  this  hoolc  prove  too  Ions  for  a 
year's  worls,  it  is  suggested  tliat  Cliapters  XVI.,  XVII.  and  XVIII.  be 
omitted,  valuable  though  they  may  be  as  affording  exercises  in 
accurate  geometrical  construction. 

Exercises. 

1 .  Draw  two  lines  making  an  angle  of  50°  with  one  another,  and 
describe  three  circles  touching  both  lines. 

2.  Two  lines  make  an  angle  of  70°  with  one  another.  Describe  a 
circle  of  radius  1^  in.  touching  both  of  them.  (Draw  a  perpendicular 
to  either  of  the  lines,  of  length  1^  in.,  and  through  its  end  draw  a 
line  parallel  to  the  line  on  which  the  perpendicular  stands,  producing 
this  parallel  until  it  meets  the  bisecting  line.) 


EXEECISES.  109 

3.  Two  lines  make  an  angle  of  40°  with  one  another.  Describe  a 
circle  touching  both  of  them  ;  also  a  second  circle  touching  the 
previous  circle  and  the  two  lines.  (At  point  where  first  circle  cuts 
bisecting  line,  draw  a  line  making  an  angle  of  55°  or  35°  with  it, 
according  to  cutting  point  selected. ) 

4.  Describe  a  triangle  with  angles  30°,  60°  and  90°,  and  hypotenuse 
3  in. ,  and  in  it  inscribe  a  circle. 

5.  Describe  a  triangle  with  angles  30°,  60°  and  90°,  and  hypotenuse 
6  in.,  and  in  it  inscribe  a  circle.  Compare  the  length  of  the  radius  of 
this  circle  with  length  of  the  radius  of  circle  in  previous  question. 

6.  Describe  a  triangle  with  sides  76,  68  and  44  millimetres,  and  in 
it  inscribe  a  circle. 

7.  In  the  case  of  the  triangle  of  the  previous  question,  describe 
circles  touching  each  side  and  the  other  two  sides  produced. 

8.  Having  obtained  the  four  circles  of  the  two  previous  questions, 
through  what  points  do  the  lines  joining  any  two  centres  pass? 
What  position  does  the  line  joining  any  two  centres  occupy  with 
respect  to  the  line  joining  the  other  two  centres  ?  Apply  tests  in 
both  cases. 

9.  Two  parallel  lines  are  1^  in.  apart,  and  a  third  line  cuts^em  at 
an  angle  of  60°.  Describe  all  the  circles  you  can,  each  touching  the 
three  lines.     What  is  the  length  of  the  radius  ? 

10.  In  the  previous  question,  what  is  the  figure  formed  by  joining 
the  centres  to  the  points  where  the  parallels  are  cut  by  the  third 
line  ?     Apply  test. 

11.  Is  there  any  position  which  three  lines  can  occupy,  such  that  no 
circle  can  be  described  touching  all  ? 

12.  Describe  an  equilateral  triangle  with  side  2  in.,  and  in  it  in- 
scribe a  circle.     Express  with  exactness  the  radius  of  this  circle. 

13.  Describe  also  a  circle  about  the  triangle  of  the  previous  ques- 
tion, and  express  with  exactness  its  radius. 

14.  Construct  a  triangle  with  sides  40,  45  and  50  millimetres,  and 
about  it  describe  a  circle. 

15.  Construct  a  triangle  with  sides  80,  90  and  100  millimetres,  and 
about  it  describe  a  circle.  Compare  the  length  of  radius  of  this 
circle  with  that  of  circle  in  previous  question. 


110  Geometry. 

16.  Is  there  any  position  which  three  points  can  occupy  with 
respect  to  one  another,  such  that  a  circle  cannot  be  described  to  pass 
through  all  ? 

17.  ABCD  is  a  quadrilateral;  A  =  85°,  B  =  80°,  C  =  95° ;  AB=60  and 
BC  =  80  millimetres.  Construct  the  quadrilateral  and  describe  a 
circle  about  it. 

18.  AB  (  =  3  in.)  and  CD  (=2  in.)  are  parallel  and  1  in.  apart.  A 
line  at  right  angles  to  one  and  through  its  bisection  passes  also 
through  the  bisection  of  the  other.  Describe  a  circle  to  pass  through 
A,  B,  C,  D. 

19.  A  line  AB  is  3  in.  long.  Describe  a  circle  of  radius  3  in.  to 
touch  AB  at  A.  Describe  a  second  circle  to  touch  the  previous  one 
and  also  AB  at  B. 

20.  From  the  fact  that  two  tangents  from  the  same  point  to  a  circle 
are  equal,  what  relation  can  you  establish  between  the  sums  of  the 
opposite  sides  of  a  quadrilateral  whose  sides  touch  a  circle  ? 

21.  Construct  a  quadrilateral  whose  sides  are  40,  30,  50  and  60 
millimetres,  and  inscribe  a  circle  in  it. 


CHAPTBR  XVI. 


Squares  and  Circles  In  and  About  Circles  and 
Squares. 

1.  To  inscribe  a  square  in  a  circle,  draw  two 
diameters  at  right  angles  to  one 
another  and  join  their  extrem- 
ities. The  construction  being 
accurately  made,  the  set-square 
will  show  that  the  angles  A,  B, 


C,  D  are  all  right  angles  j  and 
the  equality  of  the  sides  AB, 
BC,  .  .  .  may  be  proved  by 
using  the  dividers. 

Of  course  the  evident  equality  of  the  triangles  AOB, 
BOC,  .  .  .  proves  the  equality  of  the  sides,  and  the 
angles  ABC,  BCD  .  .  .  are  all  right  angles,  because 
they  are  angles  in  semicircles. 

Inscribe  a  square  in  a  circle  of  radius  40  millimetres. 
Test  the  accuracy  of  your  construction  by  examining, 
with  the  dividers,  the  equality  of  the  sides. 

Inscribe  a  rectangle  (which  is  not  also  a  square)  in 
a  circle.  Test  the  accuracy  of  your  construction  by 
examining,  with  the  set-square,  whether  the  angles  are 
all  right  angles. 

In  a  circle  whose  radius  is  3  inches,  inscribe  a  rect- 
angle, one  of  whose  sides  is  1  inch.     With  instruments 

111 


112 


Geometry. 


o 


test  the  success  of  your  construction, — the  equality  of 
opposite  sides,  the  parallelism  of  opposite  sides,  the 
right-angledness  of  the  figure. 

2.  To  describe  a  square  about  a  given  circle, 

draw  two  diameters  at  right 
angles  to  each  other,  and  through 
the  ends  of  each  diameter  draw 
lines  parallel  to  the  other.  The 
construction  being  accurately 
made,  the  set-square  will  show 
that  the  angles  of  E,  F,  G,  H  are 
all  right  angles,  and  the  equality 
of  the  sides  EF,  FG,  .  .  may  be 
proved  by  using  the  dividers. 

Evidently  the  figures  AOCE,  AODF,  ....  are  equal 
squares,  whence  we  readily  prove  that  the  sides  of 
EFGH  are  all  equal  j  and  its  angles  are  right  angles. 

Describe  a  square  about  a  circle  whose  radius  is  30 
millimeters.  Test  the  accuracy  of  your  construction 
by  finding  whether  the  sides  are  equal,  using  the 
dividers ;  and  use  the  set-square  to  determine  whether 
the  angles  are  right  angles. 

Describe  a  square  about  a  circle  whose  radius  is  IJ 
inches.  As  in  the  previous  question,  test  the  accuracy 
of  your  construction. 

Draw  two  diameters  in  a  circle  not  at  right  angles 
to  each  other,  and  draw  tangents  at  their  extremities. 
Determine  the  nature  of  the  figure  formed  by  the 
tangents  by  measuring  the  lengths  of  its  sides. 

3.  To  inscribe  a  circle  in  a  given  square,  draw 


Squakes  and  Circles. 


113 


X 


portions  of  the  diagonals  of  the 
square,  so  that  they  intersect, 
as  at  E.  Draw  EF  perpendicular 
to  one  of  the  sides.  With  EF 
as  radius,  describe  a  circle.  If 
the  construction  has  been  ac- 
curate the  circle  will  touch  the 
sides  of  the  square. 

By  drawing  the  complete  diag- 
onals it  may  readily   be  shown,   from   the  equality  of 
such  triangles   as  EFD,   EGD,  that  the   perpendiculars 
from  E  on  the  sides  are  equal. 

Describe  a  square  with  side  of  4  inches,  and  in  it 
inscribe  a  circle.  Show,  by  measurement  with  dividers 
and  set-square,  that  the  lines  joining  the  points  of 
contact  form  a  square.  Show  that  the  sides  of  this 
are  perpendicular  to  the  diagonals  of  the  original 
square. 

Inscribe  a  circle  in  the  second  square  of  the  pre- 
ceeding  question. 

Inscribe  a  circle  in  a  rhombus,  each  of  whose  sides 
is  4  inches,  and  one  of  whose  angles  is  60°. 

4.  To  describe  a  circle  about  a  given  square, 

draw  portions  of  the  diagonals  so 
that  they  intersect.  Then,  plac- 
ing the  sharp  point  of  the  com- 
passes at  E,  where  the  diagonals 
intersect,  and  the  pencil  point 
on  any  one  of  the  angles,  and 
describing  a  circle,  it -will  pass 
through  the  other  angular  points 
of  the  square. 


114  GrEOMETRY. 

The  lines  from  E  to  the  angles  are  equal  if  the 
square  has  been  accurately  constructed  and  the  diagonals 
accurately  drawn  j  for  the  diagonals  of  all  parallelo- 
grams bisect  each  other,  and  the  diagonals  of  a  square 
are  equal. 

Construct  a  square  whose  side  is  80  millimetres,  and 
about  it  describe  a  circle. 

Construct  a  square  whose  side  is  40  millimetres,  and 
about  it  describe  a  circle. 

At  the  angular  points  of  the  square  in  the  pre- 
ceding question  draw  tangents  to  the  circle,  and,  bj 
measurement  with  the  dividers  and  set-square,  show 
that  the  tangents  form  a  square. 

About  the  square  formed  by  the  tangents  in  the 
preceding  question  describe  a  circle. 

The  sides  of  a  rectangle  are  80  and  35  millimetres. 
Describe  a  circle  about  it. 

Starting  with  a  square  whose  side  is  100  millimetres, 
inscribe  a  circle  in  it,  then  a  square  within  this  circle, 
a  circle  within  the  last  square,  etc. 

With   the   angular    points   of  a    square    as  centres, 

describe  four   circles,  such   that  each  touches   two  of 

the    others.     Describe    a    circle  to    touch    these    four 
circles. 

If  ABCD  be  a  square,  and  from  AB,  BC,  CD  and  DA 
equal  lengths  AE,  BF,  CG,  DH  be  cut,  what  is  the 
figure  EFGH  ? 


Exercises.  115 


Exercises. 

1.  Inscribe  a  square  in  a  circle  of  radius  f  in.  Test  accuracy  of 
construction. 

2.  Inscribe  a  square  in  a  circle  of  radius  1^  in.  Test  accuracy  of 
construction.  Compare  length  of  side  of  square  with  that  of  side  of 
square  in  previous  question. 

Compare  area  of  square  with  that  of  square  in  previous  question. 

3.  Describe  a  circle  of  radius  If  in.  In  it  draw  two  diameters 
making  an  angle  of  30°  with  one  another,  and  join  their  extremities. 
What  is  the  resulting  quadrilateral  ?    Apply  tests. 

4.  Describe  a  circle  of  radius  30  millimetres,  and  in  it  construct  a 
rectangle  one  of  whose  sides  is  25  millimetres.  Test  accuracy  of 
construction. 

5.  Describe  a  circle  of  radius  60  millimetres,  and  in  it  construct  a 
rectangle  one  of  whose  sides  is  50  millimetres.  Test  accuracy  of  con- 
struction. 

Compare  the  length  of  the  longer  side  of  this  rectangle  with  the 
length  of  the  longer  side  of  the  rectangle  in  the  preceding  question. 
How  are  the  areas  of  the  rectangles  related  ? 

6.  Describe  a  circle  of  radius  |  in.,  and  about  it  describe  a  square. 
Test  accuracy  of  construction.  • 

7.  Describe  a  circle  of  radius  35  millimetres,  and  both  in  and  about 
it  construct  squares. 

8.  What  ratio  always  exists  between  the  sides  of  squares  about  and 
in  the  same  circle  ?     What  ratio  between  their  areas  ? 

9.  Draw  two  diameters  of  a  circle  (radius  1  in.)  at  an  angle  of  30° 
to  one  another,  and  at  their  ends  draw  tangents.  What  is  the 
resulting  quadrilateral  about  the  circle  ?    Apply  test. 

10.  About  a  circle  of  radius  35  millimetres  construct  a  rhombus 
with  angles  60°  and  120°.  Test  accuracy  of  construction.  Show  that 
the  length  of  each  side  must  be  yf-  millimetres. 

11.  Why  is  it  that  a  rectangle  or  parallelogram  about  a  circle  must 
always  be  a  square  or  rhombus  ? 

12.  About  a  circle  of  radius  1 J  in.  construct  a  rhombus  with  one 
angle  three  times  the  other.     What  is  the  length  of  the  sides  ? 


116  GrEOMETKY. 

13.  Construct  a  square  with  side  2  in.,  and  in  it  inscribe  a  circle. 
Join  points  of  contact,  and  show  by  tests  that  the  resulting  figure  is 
a  square.     What  is  its  side  ? 

14.  Construct  a  rhombus  with  sides  50  millimetres  in  length  and 
angles  75°  and  105°,  and  describe  a  circle  touching  the  sides. 

15.  Construct  a  rhombus  with  diagonals  of  60  and  80  millimetres, 
and  in  it  inscribe  a  circle.  Measure  length  of  radius,  and  test 
accuracy  of  measurement  by  calculation. 

16.  Construct  a  square  with  side  of  2  in.,  and  about  it  describe  a 
circle.  At  the  angular  points  of  the  square  draw  tangents  to  the 
circle,  and  by  tests  show  that  the  resulting  figure  is  a  square. 

17.  Construct  a  rectangle  with  sides  30  and  40  millimetres,  and 
about  it  describe  a  circle.  Measure  radius  of  circle,  and  test  accuracy 
of  measurement  by  calculation. 

18.  Construct  a  rectangle  such  that  when  a  circle  is  described 
about  it,  and  tangents  drawn  at  the  angular  points,  the  resulting 
rhombus  shall  have  angles  of  60°  and  120°. 

19.  Beginning  with  a  circle  of  radius  50  millimetres,  inscribe  a 
square  in  it,  then  a  circle  within  the  square,  and  finally  a  square 
within  this  latter  circle.     Test  the  accuracy  of  the  final  square. 

What  are  the  lengths  of  the  sides  of  the  squares,  and  the  length 
of  the  radius  of  the  second  circle  ? 

20.  About  a  circle  of  radius  1|  in.  describe  a  quadrilateral  with 
angles  60°,  150°,  110°,  40°. 

Can  you  describe  about  a  circle  a  quadrilateral  equiangular  to 
any  given  quadrilateral  ? 


CHAPTER  XVII. 


Regrular  Polygrons. 

1.  A  polygon  is  a  rectilineal  figure  contained  by 
more  than  four  straight  sides. 

A  pentagon   is   a  figure   of  5   sides. 
hexagon         "         "  6     '' 

heptagon        "         ''         7     '' 
octagon  ''         '^         8     " 

decagon  ^^         "       10     " 

dodecagon     ''         "        12     '^ 
quindecagon "         ^'        15     ^' 
A  polygon  is  said  to  be  regular  when  all  its  sides 
are  equal,  and  also  its  angles  equal. 

2.  The  angles  at  any  point,  for  example,  at  the  cen- 
tre of  a  circle,  make  up  360°.  We  can  divide  this 
interval,  by  means  of  the  protractor,  into  a  number,  5, 


6,  8, 


.  ,  of  equal  angles.     If  we  prolong  the  sides 


of  these  angles  until  they  intersect  the  circumference 
of  the  circle,  and  join  the  successive  points  of  inter- 
section, we  have  a  regular  polygon  of  5,  6,  8,  .  .  . 
.  .  .  .  sides,  as  the  case  may  be. 

3.  To  describe  a  regular  pentagon  in  a  circle  : 

A  pentagon  having  five  sides, 
the  angle  subtended  at  the  cen- 
tre of  the  circle  by  the  side  of 
a  regular  pentagon  inscribed  in 
the  circle,  will  be  i  of  360°  =  72°. 
Using  then  the  protractor,  or 
adjusting  the  bevel  to  an  angle 
of  72°,  lay  off  at  the  centre  5 
angles,  each  of  this  magnitude. 
Produce  the  sides  of  the  angles  to  meet  the  circumference, 

117 


118 


Geometky. 


at 


and  join  the  succeeding  points  of  intersection.  The 
construction  being  accurately  made,  the  bevel  will 
show  the  equality  of  the  angles  ABC,  BCD,  .  .  .  , 
and  the  dividers  will  show  the  equality  of  the  sides 
AB,  BC 

Of  course,  the  evident  equality  of  the  isosceles  tri- 
angles OAB,  OBC,  .  .  .  ,  proves  the  equality  of  the 
sides  and  angles  of  the  pentagon. 

The  angle  at  the  vertex  of  each  isosceles  triangle  in 
the  figure  being  72°,  each  angle  at  the  base  must  be 
54°j  and  therefore  each  of  the  angles  (ABC,  BCD,  .  .  .  ) 
of  a  regular  pentagon  is  108°. 

4.  If  tangents  to  the  circle  be  drawn 
the  angular  points  of  the 
pentagon  ABCDE,  the  tan- 
gents form  another  regular 
pentagon,  which  is  said 
to  be  about  the  circle. 

The  equality  of  the  sides  FG, 
GH,  .  .  .  may  be  tested  with 
the  dividers,  and  the  equality 
of  the  angles  FGH,  GHK,  .  .  . 
with  the  bevel. 

5.  If  we  wish  to  construct  on  a  given  straight 
line  (AB),  as  side,  a  regular 
pentagon,  at  the  points  A 
and  B,  with  the  protractor  we 
mark  off  angles  BAE,  ABC  of 
108°,  and  with  the  dividers 
make  BC  and  AE,  each  equal 
to  AB.  At  C  we  again  make 
an  angle  BCD  of  108°,  and 
mark    off    CD    equal    to    AB. 


Exercises.  119 

Joining  E  and  D,  we  have  a  regular  pentagon  ABCDE. 
Using  the  bevel,  we  shall  find  that  the  angles  at  E  and 
D  are  equal  to  the  three  other  angles,  and  the  dividers 
will  prove  the  side  DE  to  be  equal  to  the  other  sides. 

The  radius  of  a  circle  being  36  millimetres,  inscribe 
in  it  a  regular  pentagon.  "With  the  dividers  and  bevel 
prove  the  accuracy  of  your  construction, — that  the 
sides  and  angles  are  equal. 

Describe  also  about  the  same  circle  a  regular  penta- 
gon. With  the  dividers  and  bevel  prove  the  accuracy 
of  your  construction. 

On  a  line  of  length  2  inches,  as  side,  construct  a 
regular  pentagon.  With  instruments  prove  the  accu- 
racy of  your  construction. 

Exercises. 

1.  In  a  circle  of  radius  32  millimetres,  inscribe  a  regular  pentagon. 
Test  equality  of  sides  with  dividers,  and  equality  of  angles  with  bevel 
or  protractor. 

2.  In  a  circle  of  radius  If  in.,  inscribe  a  regular  pentagon.  Test 
accuracy  of  construction. 

3.  About  a  circle  of  radius  1^  in.,  describe  a  regular  pentagon. 
Test  accuracy  of  construction. 

4.  About  a  circle  of  radius  f  in.,  describe  a  regular  pentagon.  Test 
accuracy  of  construction. 

5.  In  the  two  preceding  questions,  where  the  radius  of  one  circle  is 
twice  that  of  the  other,  examine  the  relation  between  the  lengths  of 
all  corresponding  lines  that  can  be  drawn  in  the  two  figures, — sides, 
lines  joining  non-adjacent  angles,  segments  of  these  lines  by  their 
intersection. 

6.  Inscribe  two  regular  pentagons  in  any  two  circles  of  diflferent 
radii.  With  the  bevel  examine  the  relation  between  all  correspond- 
ing angles  that  can  be  formed  in  the  two  figures. 

7.  Describe  an  irregular  equilateral  pentagon,  each  side  being  1  in. 


120  Geometry. 

8.  About  a  circle  of  radius  1|  in.,  describe  a  pentagon  with  angles 
80°,  110°,  145°,  70°,  and  135°. 

9.  Describe  a  regular  pentagon  with  side  of  1  in.  Test  accuracy 
of  construction. 

10.  Describe  a  regular  pentagon  with  side  of  2  in.  Test  accuracy 
of  construction. 

11.  In  the  two  preceding  questions,  what  is  the  relation  between 
the  radii  of  the  two  circles  about  the  pentagons  ?  • 

12.  Hence  if  you  have  in  a  circle  (radius  OA)  a  regular  pentagon 
with  side  30  millimetres,  how  many  times  OA  should  you  make  the 
radius  of  a  second  circle,  that  the  side  of  a  regular  pentagon  in  it  may 
be  45  millimetres  ? 

13.  ABCDE  being  a  regular  pentagon,  what  sort  of  triangles  are 
ACD,  and  ABC  ?  What  are  the  magnitudes  of  the  angles  CAD, 
ACD,  CBD  ? 

14.  In  the  figure  of  the  preceding  question,  join  each  angle  to  the 
other  angles.  Is  the  pentagon  thus  obtained,  in  the  centre  of  the 
figure,  regular  ?  Apply  tests.  Measure  each  angle  of  the  figure, 
formed  by  intersecting  lines,  and  assign  to  it  its  magnitude  in 
degrees. 

15.  Since  the  side  of  a  regular  pentagon  subtends  an  angle  of  72° 
at  the  centre  of  the  circle  about  it,  what  angle  should  a  side  subtend 
at  the  circumference  ?  Hence  assign  to  each  angle  at  circumference 
in  question  14,  its  proper  magnitude,  and  deduce  values  of  all  other 
angles  in  the  figure. 

16.  In  the  figure  of  question  14,  indicate  all  lines  that  are  equal  to 
one  another  ;  also  all  triangles  that  are  isosceles. 

17.  In  the  same  figure  erase  the  circumference,  and  sides  of  the 
pentagon,  so  obtaining  a  star-shaped  figure.  Show  how  such  a  figure 
(called  a  pentagram)  could  be  described  without  taking  the  pencil 
from  the  paper. 

18.  Without  describing  a  circle,  construct  a  pentagram,  the  line 
corresponding  to  AC  being  3  in.  Test  accuracy  of  construction  by 
determining  lengths  AB,  BC,  .  .   .    ,    and  angles  ABC,  BCD,    .... 

19.  In  the  figure  of  question  14,  how  many  rhombuses  are  there? 

20.  With  respect  to  how  many  lines  is  a  regular  pentagon  sym- 
metrical ?     Has  it  central  symmetry  ? 


CHAPTER  XVIII. 

Reg"Ular  Polyg"Ons     (Continued).     , 

1.  To  inscribe  a  regular  hexagon  in  a  circle; 

A  hexagon  having  six  sides, 
the  angle  subtended  at  the  cen- 
tre of  the  circle  by  the  side  of 
a  regular  hexagon  inscribed  in 
the  circle,  will  be  i  of  360°  =  60°. 
Using  then  the  protractor,  or 
adjusting  the  bevel  to  an  angle 
of  60°,  lay  off  at  the  centre 
two  angles  of  60°.  Produce 
the  three  sides  of  these  angles  both  ways  to  the  cir- 
cumference, and  join  the  succeeding  points  of  intersec- 
tion. The  construction  being  accurately  made,  the 
bevel  will  show  the  equality  of  the  angles  ABC,  BCD, 
.  .  .  ,  and  the  dividers  will  show  the  equality  of  the 
sides  AB,  BC, 

Since,  however,  each  of  the  triangles  in  the  figure 
is  equilateral,  having  its  sides  equal  to  the  radius, 
the  sides  of  the  hexagon  are  equal  to  the  radius  of 
the  circle.  Hence  the  easiest  way  to  describe  a  hexa- 
gon in  a  circle  is  to  measure  off,  with  the  dividers, 
six  chords  in  succession,  each  equal  to  the  radius. 

Evidently  the  angle  of  a  regular  hexagon  is  120°. 

121 


122 


Geometry. 


2.  If  tangents  to  the  circle  be  drawn  at 
the  angular  points  of  the 
hexagon  ABCDEF,  the  tan- 
gents form  another  hexa- 
gon, which  is  said  to  be 
about  the  circle.  The  equality 
of  the  sides  GH,  HK,  ....  may 
be  tested  with  the  dividers,  and  k 
the  equality  of  the  angles  GHK, 
HKL,  .  .  .  with  the  bevel. 


3.  If  we  wish  to  construct  a  regular  hexagon  with 
sides  of  given  length,  we  describe  a  circle  with  radius 
of  this  length,  and  in  it  inscribe  a  regular  hexagon  as 
in  §  1. 

4.  To  inscribe  a  regular  octagon  in  a  circle : 

We  may  construct  at  the 
centre  eight  angles,  each  of 
45°,  and  join  the  ends  of 
consecutive  radii  bounding 
these  angles  j  or,  perhaps 
more  conveniently,  we  may 
proceed  as  follows  :  Draw  two 
diameters  at  right  angles  to 
one  another  and  join  their 
extremities.  We  thus  have  a 
square  in  the  •  circle.  Through  the  centre,  using 
parallel  rulers,  draw  diameters  parallel  to  the  sides  of 
the  square.  The  quadrants  are  thus  bisected,  and  we 
get  eight  equal  angles  at  the  centre.  Joining  ends  of 
the  successive  radii  which  bound  these  angles,  we 
have  an  octagon  inscribed  in  the  circle.     The  accuracy 


Eegulak  Polygons. 


123 


of    the    construction    may    be    tested    by  using    the 

dividers    to    determine    whether    the   sides  are    equal, 

and   the  bevel  to   determine   whether   the  angles    are 
equal. 

Each  of  the  angles  at  the  centre  is  45°.  Hence 
each  of  the  angles  at  the  base  of  any  of  the  isosceles 
triangles,  OAB,  OBC,  ...  is  67J°,  and  the  angle  of  a 
regular  octagon  is  135°. 

5.  If  tangents  be  drawn  at  the  angular  points 
of  the  octagon  ABCDEFGH,  the  tangents  form 
another  regular  octagon  which  is  said  to  be 
about  the  circle. 

6.  To  describe  a  regular  octagon  with  side, 
AB,  of  given  length  we  may  proceed  as  follows: 

Construct  the  angle  ABC  of  135°,  and  make  BC  =  AB. 
Bisect  AB  and  BC  in  K  and  L,  and  draw  KO,  LO  perpen- 
dicular to  AB  and  BC.  With  0  as 
centre,  and  radius  OA,  OB  or  OC 
describe  a  circle.  On  this  lay 
off  with  the  dividers  six  chords 
equal  to  AB  or  BC,  beginning 
at  the  point  C  or  A.  That  the 
rest  of  the  circle  is  exactly 
taken  up  with  six  such  chords 
affords  a  test  of  the  accuracy 
with  which  the  angle  ABC  (135°)  is  constructed,  AB 
and  BC  are  bisected,  and  the  perpendiculars  KO  and 
LO  are  drawn. 

7.  The  pupil  may  continue  these  exercises,  con- 
structing regular  decagons,  dodecagons,  etc.,  in  a  way 
quite  analogous  to  the  preceding  constructions. 


124  Geometry. 

The  radius  of  a  circle  being  1|  in.,  inscribe  in  it  a 
regular  hexagon.  Test  the  accuracy  of  your  construc- 
tion by  testing  the  equality  of  all  the  angles. 

Describe  a  regular  hexagon  about  the  circle  in  the 
preceding  question,  testing  the  equality  of  sides  and 
angles  of  the  figure. 

Construct  a  regular  hexagon  with  sides  1^  in. 

Construct  a  figure  similar  to 
that  annexed,  in  which  the  outer 
circle  touches  six  smaller  ones. 

Construct  the  figure  also  so 
that  the  six  small  circles  touch 
one  another,  and  are  all  touched 
by  the  outer  (large)  and  inner 
(small)  circles.  (Radius  of  small 
circles  should  be  one-third  radius  of  large  circle.) 

Describe  a  regular  octagon  in  a  circle  whose  radius 
is  43  millimetres.  Test  the  accuracy  of  your  con- 
struction by  testing  the  equality  of  the  sides  (using 
dividers),  and  by  examining  whether  each  of  the  angles 
of  the  octagon  is  135°. 

Construct  a  regular  octagon  whose  side  is  2  inches. 
Examine  the  accuracy  of  your  construction  by  testing, 
with  the  dividers,  the  equality  of  the  sides,  and,  with 
the  bevel,  the  equality  of  the  angles. 

Describe  eight  circles  of  the  same  radius,  each 
touching  two  others  of  the  set,  and  the  entire  eight 
lying  within  and  being  touched  by  a  ninth  circle  of 
given  radius. 

The  general  way  of  solving  such   a  problem   as  the 


Exercises.  125 

preceding  is  as  follows:     Suppose 

the  number  of   small   circles  is  to 

be   8,    9,    .    .   .     Let   AOB   be   the 

8th,  9th,  .  .  .  ,  as  the  case  may  be, 

part   of    360°.     Bisect   the    angles 

OAB,  OBA  by   AC,   BC.     Through 

C  draw  DCE  parallel  to  AB.     Then 

evidently  DA,  DC,  EB,  EC  are  all 

equal,    and    the     circle     described 

with  D  as  centre,  and  DA  or  DC  as  radius,  will  touch 

the  cii-cle  described  with  E  as  centre,  and  EB  or  EC  as 

radius  J  and  both  circles  will  touch  the  large  one. 

Exercises. 

1.  In  a  circle  of  radius  1|  in.,  inscribe  a  regular  hexagon. 

2.  Describe  a  regular  hexagon,  the  sides  being  35  millimetres. 

3.  Describe  a  regular  hexagon  with  side  of  2  in.  Join  alternate 
angles,  so  obtaining  a  star-shaped  figure  with  six  points.  What  is 
the  six-sided  figure  at  centre  of  this?  Apply  tests.  What  are  the 
various  triangles  in  the  figure  ?     Apply  tests. 

4.  In  the  figure  of  the  preceding  question,  at  what  various  angles 
are  the  sides  of  the  hexagon  at  centre  inclined  to  any  side  of  the 
original  hexagon  ? 

5.  About  a  circle  of  radius  40  millimetres  describe  a  hexagon  with  - 
angles  90°,  100°,  110°,  130°,  140°,  150°. 

6.  A  regular  hexagon  is  described  about  a  circle  of  radius  2  in. 
Show  that  the  side  of  the  hexagon  is  -^^3-  in. 

7.  The  side  of  a  regular  hexagon  is  2  in.  What  is  the  length  of 
the  radius  of  the  circle  inscribed  in  it  ? 

8.  Inscribe  a  regular  octagon  in  a  circle  of  radius  32  millimetres. 
Test  accuracy  of  construction. 

9.  In  a  circle  of  radius  50  millimetres,  inscribe  a  regular  octagon, 
ABCDEFGH.  Join  AD,  DG,  GB,  .  .  .  .  ,  each  time  passing  over 
two  angles,  and  so  obtaining  a  star-shaped  figure  with  eight  points. 
What  is  the  figure  formed  at  centre  ?     Apply  tests. 


126  Geometry. 

10.  In  the  preceding  figure,  what  are  the  various  triangles  formed  ? 
At  what  various  angles  are  the  sides  of  the  octagon  at  centre  inclined 
to  any  side  of  the  original  octagon  ? 

11.  In  the  same  figure,  what  angles  alone  occur  ?  How  many 
rhombuses  are  there  in  the  figure  ? 

12.  Construct  a  regular  octagon  whose  side  is  35  millimetres.  Test 
the  accuracy  of  your  construction. 

13.  With  the  angular  points  of  a  regular  octagon  as  centres, 
describe  eight  circles  of  equal  radii,  so  that  each  touches  two  others  of 
the  set. 

14.  With  respect  to  how  many  lines  is  a  regular  hexagon  sym- 
metrical ?     Has  it  central  symmetry  ? 

15.  With  respect  to  how  many  lines  is  a  regular  octagon  sym- 
metrical ?  Has  it  central  symmetry  ?  Has  a  regular  heptagon 
central  symmetry  ? 

16.  In  a  circle  of  radius  37  millimetres  inscribe  a  regular  dodecagon. 

17.  What  is  the  ratio  of  the  sides  of  two  regular  hexagons,  one 
inscribed  in,  and  the  other  described  about,  the  same  circle  ? 

18.  ABCDEF  is  a  regular  hexagon.  Show  that  its  area  is  twice 
that  of  the  equilateral  triangle  ACE. 

19.  In  a  circle  the  angle  ABC  is  equal  to  the  angle  BCD.  How  are 
the  chords  AB,  CD  related  ? 

20.  An  equiangular  polygon  inscribed  in  a  circle  has  its  alternate 
sides  equal. 

21.  At  B,  a  point  on  a  circle,  construct  an  angle  ABC  of  108°  (the 
angle  of  a  regular  pentagon),  the  sides  AB,  BC  not  being  equal.  At 
C  make  BCD  of  108° ;  at  D  make  CDE  of  108°  ;  and  so  on.  Shall  we 
at  length  reach  accurately  the  point  A  ?  If  so,  after  how  many  times 
about  the  circle  ?  Has  a  regular  pentagon  been  described  ?  Can 
other  regular  pentagons  be  obtained  from  the  figure  by  producing 
lines  or  otherwise  ? 


CHAPTER  XIX. 
Similar  Triangrles. 

1.  Two  triangles  are  similar  when  the  angles  of 
one  triangle  are  equal  to  the  angles  of  the  other,  the 
sides  not  necessarily  being  equal. 

Thus  if  two  triangles  of  different  sizes  have  their 
angles  45°,  65°  and  70'',  they  are  similar. 

In  the  following  article  a  remarkable  property  of 
such  triangles  is  reached. 


2.  On  a  base  BC  of    15  millimetres   construct   a  tri- 
angle  with   sides   AB,  AC   of   20   and   25  millimetres. 

127 


128  G-EOMETKY. 

Draw  two  other  bases  BX^t  B2C2  of  lengths  30  jid 
45  millimetres.  At  B^  and  B^  make  angles  CiB^Ai, 
C2B0A2,  each  equal  to  CBA;  and  at  C^  and  C2  make 
angles  BiCiAi,B2C2A2,  each  equal  to  BCA.  It  follows 
(Ch.  III.,  4)  that  the  angles  at  A,  A^,  A2  are  equal  to 
one  another.  Hence  the  three  triangles  are  equiangu- 
lar and  similar. 

Now  measure  the  lengths  of  the  sides  of  the  tri- 
angles AiBjCi  and  AoBgCg.  If  the  constructions  have 
been  accurately  made,  we  shall  have  the  following 
numerical  values: 


BC=15 

BiCi=30 

B2C2  =45 

AB  =  20 

A,B,..40 

A2B2=60 

AC  =25 

A^^  ^50 

A2C2=75 

Then  calling  those  sides  corresponding  sides  which 
are  opposite  to  equal  angles,  we  observe  that  corres- 
ponding sides  about  equal  angles  are  proportional^  Le.y 


J_5      3(1     4  5. 

20      —      40       —       60 


AO.      60. 

5  0       —       7  5 


±5      __      AO,      _       4_5 
2  5  5  0  7  5 


3.  Again,  construct  a  triangle  ABC,  whose  base  BC 
is  24,  and  sides  AB  and  AC,  30  and  40  millimetres. 
Draw  two  other  bases  B^C^  and  B2C3  of  lengths  36 
and  60  millimetres.  At  B^  and  B2  make  angles  CiBjA^, 
C2B2A2,  each  equal  to  CBA;  and  at  Cj  and  Cg  make 
angles  BiCjAj,  B2C2A2,  each  equal  to  BCA.  It  follows 
(Ch.  III.,  4)  that  the  angles  at  A,  A^,  Ag  are  equal  to 
one  another.  Hence  the  three  triangles  are  equiangu- 
lar and  similar. 


Similar  Triangles. 


129 


Now  measure  the  lengths  of  the  feides  of  the  tri- 
angles AiB^Ci,  A2B2C2.  If  the  constrnctions  have 
been  accurately  made,  we  shall  have  the  following 
numerical  values: 


130  Geometry. 

BC=24  BiCi=36  BgC^  =   60 

AB  =  30  AiBi=45  A2B2=  75 

AC  =40  AiCi=60  AgCg^lOO 

And  we  again  find  that  corresponding  sides  about 
equal  angles  are  proportional,  i.e,, 

2.A    3.^    6_0       - 

30     —     45     —     75 

3_0  _  4:  J.  _  7  5 

40  —  60  —  '100 

2  4  3. 6.  60 

40  —  60  —  100 

4.  The  pupil  may  repeat  this  experiment  with  equi- 
angular triangles,  and,  the  constructions  being  accurate- 
ly made,  he  will  always  reach  the  same  conclusion  as 
to  the  proportionality  of  the  corresponding  sides  about 
equal  angles. 

(The  easiest  way  to  secure  the  equality  of  the  angles 
is  to  place  with  the  parallel  rulers  B^Ci  parallel  to 
BC,  and  then  with  the  same  rulers  draw  B^A^  parallel 
to  BA,  and  CiA^  parallel  to  CA.) 

The  result  of  these  observations  may  be  stated  thus : 

The  sides  about  the  equal  angles  of  equi- 
angular triangles  are  proportionals ;  and  cor- 
responding sides,  i.e.,  those  which  are  opposite 
to  equal  angles,  are  the  antecedents  or  con- 
sequents of  the  ratios. 

(Note :  In  the  ratio  a :  b,  a  is  called  the  antecedent, 
and  b  the  consequent.) 

This  is  the  most  important  proposition  in  Geometry: 
indeed,  one  of  tiie  most  important  results  of  all 
science.  Through  it,  in  effect,  all  measurements  are 
made  when  we  cannot  actually  go  over  the  distance 
to  be  measured  with  a  rule,  a  surveyor's  chain,  or 
other  measuring  instrument. 


Similar  Triangles. 


131 


5.  The  result  reached  in   the   preceding  article  may 
be  demonstrated  more  generally  as  follows : 

Let  ABC,  AjBiCi  be  similar  triangles,  and  let  them 
be  placed  so  that 
AB  rests  on  A^B^, 
and  AC  on  A^C^,  as 
in  the  figure.  Then 
BC  is  parallel  to 
BjCj.  Suppose  AB 
and  AjBj  commen- 
surable, and  let  AB 
contain  n  units,  and 
AiBi  contain  n-^ 
units.      Suppose 

AjB^  divided  into  its  units,  and  through  the  points  of 
division  draw  lines  parallel  to  BC  or  B^C^.  Evidently 
the  divisions  of  A^C^  are  all  equal  to  one  another, 
though  not  necessarily  equal  to  those  of  A^B^.  Then 
also  AC  contains  7i  parts  equal  to  AE,  as  AB  contains 


71  parts   equal   to    AD;    and    A^C^    contains 


parts 


equal  to  AE,  as  A^B^  contains  n^  parts   equal   to  AD, 
Hence 

^  n  AC 

AjBi    ~   n^    ~  A^Ci 

In  like  manner  the  proportionality  of  the  sides  about 
the  other  equal  angles  may  be  shown. 

6.  On  the  other  hand,  if  the  lengths  of  the  sides  of 
one  triangle  may  be  obtained  from  the  lengths  of  the 
sides  of  another  by  multiplying  or  dividing  each  by 
the  same  number ;  that  is,  if  the  sides  of  two  triangles, 
taken  in  order,  are  proportional,  what  relation  exists 
between  the  angles  of  the  two  triangles? 


132  GrEOMETRY. 

Construct  and  examine  the  following  triangles^  and 
see  if  you  can  supply  an  answer  to  the  question: 

(1)  Sides  20,  30,  40,  and  40,  60,  80  millimetres. 

(2)  Sides  1,  IJ,  IJ,  and  IJ,  2^,  2|  inches. 

(3)  Sides  24,  36,  40,  and  42,  63,  70  millimetres. 

Exercises. 

1.  The  sides  of  two  triangles  are  20,  30,  40,  and  40,  60,  80  milli- 
metres, respectively.  Construct  them,  and,  using  the  bevel,  show 
that  they  are  equiangular. 

2.  The  sides  of  two  triangles  are  20,  30,  40  and  30,  45,  60  milli- 
metres, respectively.  Construct  them,  and  show  that  they  are  equi- 
angular. 

3.  The  bases  of  two  triangles  are  35  and  60  millimetres,  and  the 
angles  adjacent  to  each  base  are  75°  and  70°.  Construct  the  triangles, 
and  show  that  corresponding  sides  are  as  35  :  60. 

4.  Construct  two  triangles  of  different  sizes  with  angles  35°,  45°  and 
100°.  On  a  line  AB  lay  off  lines  equal  to  the  sides  of  one  triangle  ; 
and  on  another  line  AC  lay  off  lines  equal  to  the  sides  of  the  other 
triangle.  Let  the  ends  of  corresponding  lengths  on  AB,  AC  be 
joined.  What  position  do  these  joining  lines  occupy  with  respect 
to  each  other  ?    Apply  test.     What  is  the  inference  ? 

5.  The  angles  of  two  triangles  are  60°,  75°  and  45°.  Construct  the 
triangles,  and,  after  the  manner  suggested  in  question  4,  test  the  pro- 
portionality of  the  sides. 

6.  The  angles  of  two  triangles  are  110°,  30°  and  40°,  and  the  sides 
opposite  angle  of  30°  in  each  are  40  and  55  millimetres.  Construct 
the  triangles,  and,  after  the  manner  suggested  in  question  4,  test  the 
proportionality  of  the  sides. 

7.  The  angles  at  the  vertices  of  two  triangles  are  both  36°.  The 
sides  adjacent  to  the  vertex  of  one  triangle  are  1|  in.  and  2  in.,  and 
adjacent  to  the  vertex  of  the  other  2|  in.  and  3  in.  Construct  the 
triangles.  Show  by  measurement  that  angles  opposite  corresponding 
sides  are  equal,  and  that  the  remaining  sides  are  in  ratio  1  :  1^. 


Exercises.  133 

8.  The  angles  at  the  vertices  of  two  triangles  are  both  67°,  and  the 
sides  about  these  angles  are  40,  60  and  44,  66  millimetres.  Con- 
struct the  triangles.  Show  by  measurement  that  triangles  are  equi- 
angular, and  that  the  remaining  sides  are  as  10  :  11. 

9.  Construct  an  angle  BAG  of  39°,  and  from  P  in  AC  draw  PN  per- 
pendicular to  AB.  Measure  the  lengths  of  AP,  AN,  PN  in  milli- 
metres, and  find  the  numerical  values  to  two  places  of  decimals  of  the 
ratios 

PN       AN  ^      PN 

AP'      AP      ^""^      AN' 

10.  In  the  preceding  question,  keeping  to  the  angle  of  39°,  take 
the  point  P  in  different  positions  on  AC,  drop  the  perpendicular 
PN,  for  each  position  of  P  repeat  the  measurements  and  calculate  to 
two  decimal  places  the  values  of  the  preceding  ratios.  Compare 
values  with  those  already  obtained. 

11.  Keeping  to  same  angle  39°,  take  the  point  P  in  AB  and  drop 
PN  perpendicular  on  AC.     Again  calculate  these  ratios. 

State  your  conclusion  as  to  the  values  of  these  ratios, — perp.  to 
hyp.  ;  base  to  hyp.  ;  perp.  to  base — so  far  as  the  angle  39°  is  concerned. 

12.  BC  of  a  right-angled  triangle  ABC  (C  =  90°)  is  found  to  be 
748  ft. ,  and  the  angle  ABC  is  39°.  Use  the  results  of  the  three  pre- 
ceding questions  to  find  approximately  the  lengths  of  AC  and  AB  in 

feet. 


chapt:^r  XX. 

Similar  Triangrles.    (Continued). 

1.  In   the   annexed   figure   the    triangles    ABC,  ADE 
are    similar.       Suppose    the    values  of    the    lines    are 


AD  =  59,  AB  =  32,  BC-24,    and    that   DE    is   unknown. 
The  property  of  similar  triangles  gives 

DE  ^  24 

59         32 
24 


DE 


32 


X  59  =  441 


2.  If  level  ground  can  be  found  extending  out  from 
the  base  of  a  tree,  or  other  vertical  object,  its  height 
may  be  found  as  foUows: 

Let  two  rods,  AB  and  CD,  be  placed  upright  in  the 
ground,  at  such  distance  apart  that  the  eye  sees  the 
tops  (B  and  D)  of  the  rods  and  the  top  (F)  of  the 
tree  in  the  same  straight  line. 

The  heights  of  the  rods  being  measured,  their  dif- 
ference DG  is  known.  Let  also  the  lengths  AC  (i.e.,  BG) 
and  CE  {i.e.,  GH)  be  measured. 

134 


Similar  Triangles. 


135 


A         C 

Suppose  AC  =  BG  =  11,  CE==GH  =  43, 
AB  =  13,  CD  =  20. 

Theu  by  similar  triangles  BGD,  BHF 

^^      =  1      HF    =    1    X    54  = 

43  +  11       11 ;  11 


34 


4 
11 


Then  height  of  object,  EF  =  34A-  +  13  =  47-A- 

3.  Suppose  we  wish  to  find  the  distance  of  an  object 
B  from  A,  without  going  over  the  distance  AB  with  a 
surveyor's  chain  or  other  instrument  for  measuring. 

Measure  a  base  line,  AC,  of,  say,  250 
feet,  and  note  the  angles  CAB,  ACB. 
Then,  on  paper,  construct  a  triangle 
A^BjCj,  equiangular  to  ABC,  but  with 
a  base  line  A^Cj  of,  say,  1  foot.  Measure 
the  length,  in  feet,  of  A^Bi.  The  line 
AB  will  be  250  times  the  length  of 
A,B,. 

This  example  embodies  the  principle 
of  the  range-finder,  so  much  used  in 
military  and  naval  operations. 


136 


Geometry. 


4.  Diagrams  such,  as  the 
following  should  be  con- 
structed with  accuracy^ 
where  DE  is  parallel  to 
BC,  and  therefore  the  tri- 
angles ABC  and  ADE  simi- 
lar. AB,  BC  and  AD 
should  then  be  measured 
the  proportion 

DE        BC 


and    DE    calculated    from 


or 


DE 


AD, 


BC 
AD   -   AB'  "^ AB 

and  the  accuracy  of  the  construction,  measurements 
and  calculation  tested  by  measuring  DE  with  the 
dividers  and  scale. 

5.  The  proportionality  of  the  sides  of  similar  tri- 
angles may  be  employed  to  reduce  or  enlarge  a  figure 
to  any  scale. 


Suppose  we  wish  to  obtain  a  figure  the  same  shape 
as  ABC  .  .  .  ,  but  with  linear  dimensions  half  those  of 
ABC  .  .  .  Take  a  line  OA'A,  with  OA'  =  A'A.  From 
0  draw  a  number  of  lines  OA,  OB,  .  .  .  With  the 
parallel  rulers  obtain  B',  through  A'B'  being  parallel 
to  AB;    also   C,   through  A'C   being  parallel  to  AC; 


Exercises.  137 

also  D',  through  A'D'  being  parallel  to  AD;  and  so 
on.  Then,  with  the  judgment  of  the  eye,  fill  in  the 
contour  between  A'  and  B'  similarly  to  that  between 
A  and  B ;  between  B'  and  C  similarly  to  that  between 
B  and  C ;  and  so  on.  Any  two  points  in  the  larger 
figure  should  be  just  twice  as  far  apart  as  the  two 
corresponding  points  in  the  smaller,  and  this  may  be 
used  to  test  the  accuracy  of  the  drawing. 

Maps  may,  in  this  way,  be  reduced  or  enlarged,  the 
first  drawing  being  obtained  by-  using  translucent 
paper,  or  by  tracing  against  a  window  pane.  Of 
course  the  drawing  of  all  maps  is,  in  part,  a  question 
of  the  construction  of  similar  figures. 

Exercises. 

1.  Draw  any  line  AB  and  divide  it  in  the  ratio  of  7  to  8  by  draw- 
ing another  line  ACD,  inclined  to  AB  at  any  angle,  such  that  AC  =  28 
and  CD  =  32  millimetres,  completing  construction  with  parallel  rulers. 
Verify  result  by  measuring  segments  of  AB. 

2.  Divide  a  line  4  in.  long  in  the  ratio  3.4  to  4.1. 

3.  There  are  three  lines  of  lengths  27,  39  and  64  millimetres.  Con- 
struct geometrically  for  a  fourth  proportional  to  them,  and  verify 
result  by  calculation  and  measurement. 

4.  A  line  is  4J  in.  in  length.  Divide  it  into  three  parts,  such  that 
they  shall  be  to  one  another  as  7  :  8  : 9. 

5.  Draw  a  line  AB  an  inch  long.  Draw  another  line  AC  of  length 
50  millimetres,  inclined  to  former  at  any  angle.  Divide  the  inch  line 
into  tenths. 

6.  Divide  an  inch  into  twelfths. 

7.  Draw  AB,  AC,  making  an  angle  of  47°  with  one  another.  In 
either  of  them  take  a  point  P  and  drop  a  perpendicular  PN  on  the 
other.  Measure  the  lengths  of  the  sides  of  APN,  and  obtain  the 
numerical  values  of  the  following  ratios  to  two  decimal  places, — 


138  Geometry. 

PN       AN  PN 

AP'       AP       ^""^      AN- 

(Most  accurate  results  will  be  obtained  by  taking  P  at  some  dis- 
tance from  A,  and  measuring  in  millimetres.)     . 

8.  Take  P  in  other  line,  at  different  distance  from  A,  make  similar 
construction,  measure  sides  of  APN,  and  again  find,  to  two  decimal 
places,  the  values  of  the  above  ratios  for  47°. 

9.  Calling  the  side  opposite  47°  the  perpendicular,  the  side  opposite 
the  right  angle  the  hypotenuse,  and  the  remaining  side  the  base, 
whether  it  be  on  the  upper  or  lower  line,  are  the  above  ratios,  i.e., 

perp.        base  perp. 

hyp.  '       hyp.       '  base 

always  the  same  for  47°,  or  do  they  depend  on  where  the  point  P  is 
taken  ? 

10.  With  the  explanation  in  the  preceding  question,  find  the 
values  of  these  same  ratios 

perp.        base  perp. 

hyp.  '       hyp.  base  ' 

for  an  angle  of  63°,  to  two  decimal  places. 

11.  It  is  required  to  find  the  distance  of  a  point  C  from  an  object 
B  on  the  other  side  of  a  chasm.  For  this  purpose  a  line  CA  is  run  at 
right  angles  to  BC.  AC  is  found  to  be  278  feet,  and  the  angle  to  A 
to  be  47°.     What  is  the  distance  of  B  from  C  ? 

12.  In  the  preceding  question,  if  AC  be  344  feet,  and  the  angle  at 
A  be  63°,  what  is  the  distance  of  B  from  C  ?  Find  also  the  length  of 
AB. 

13.  To  find  how  far  a  distant  object  C  is  from  A,  a  base  line  AB  is 
measured  of  400  ft,  and  the  angles  at  A  and  B  are  found  to  be  75° 
and  80°.  Then  on  paper  a  line  DE  of  length  3  in,  is  drawn,  and 
angles  EDF,  DEP  are  constructed  of  75°  and  80°,  respectively, — and 
FD  is  measured  in  inches  and  fractions  of  an  inch.  What,  ap- 
proximately, is  the  length  of  CA  ? 

14.  If,  in  the  preceding  question,  AB  be  250  feet,  and  the  angles 
at  A  and  B  be  65°  and  77°,  respectively,  by  constructing  a  similar 
triangle  on  paper  and  measuring  the  sides,  determine  approximately 
the  distances  AC  and  BC. 


EXEKCISES.  139 

15.  In  triangle  ABC,  AC  =  372  feet,  A =48°,  C  =  90°.  Find  ap- 
proximately the  length  of  BC,  having  previously  found  for  48°  the 

^.    perp. 
ratio  ^ — ^-  ■ 
base 

16.  Draw  an  irregular  quadrilateral,  and  construct  another  of  same 
shape  and  with  linear  dimensions  half  those  of  former.  Verify 
equality  of  corresponding  angles,  and  ratio  of  sides  and  of  diagonals. 

17.  Draw  an  irregular  pentagon,  and  construct  another  of  same 
shape  and  with  linear  dimensions  one-third  those  of  former.  Verify 
equality  of  corresponding  angles,  and  ratio  of  sides  and  of  diagonals. 

18.  Make  an  outline  map  of  the  state  of  Michigan  with  linear  dimen- 
sions half  or  twice  those  in  map  of  United  States  given  in  your  atlas. 
Verify  correctness  by  finding  ratio  of  distance  5  between  pairs  of 
corresponding  points. 

19.  Make  a  map  of  the  Mississippi  and  Ohio  rivers  from  Quincy, 
111.,  and  Cincinnati  to  Memphis,  half  or  twice  the  size  of  that  given  in 
your  atlas.  Test  correctness  by  finding  ratio  of  distances  between 
pairs  of  corresponding  points. 

20.  Construct  a  triangle  with  sides  50,  30  and  48  millimetres. 
Bisect  the  angle  opposite  the  last  side.  In  what  ratio  are  the 
segments  into  which  this  bisecting  line  divides  this  side  ?  Does  the 
same  ratio  exist  elsewhere  in  the  figure  ? 


CHAPTER  XXI. 


L 

I 

\ 

K 

\ 

Similar  Triang-Ies.     (Continued). 

1.  Let   ABC   and   DEF  be    similar   triangles,   having 
the  base  EF  three  times  the  base  BC.     The  other  sides 
of  DEF  are   therefore    three   times  the   corresponding 
sides  of  ABC.      If  DK 
and  AG  be  the  perpen-  ^ 

dicnlars  to  the  bases, 
the  triangles  ABG  and 
1)EK  are  equiangular, 
and  therefore,  since 
DE  is  three  times  AB, 
DK  is  also  three  times 
AG.  ^ 

If  rectangles  be  con- 
structed  on   the   bases 

equal  to  the  triangles,  the  heights  of  these  rectangles 
are  half  the  heights  of  the  triangles  (Ch.  VIII.,  5). 
Hence  FN,  which  is  half  of  DK,  is  three  times  CL, 
which  is  half  of  AG. 

So  that  the  rectangle  EFNP  (which  is  equal  to  the 
triangle  DEF)  is  three  times  as  long  and  three  times 
as  high  as  the  rectangle  BCLM  (which  is  equal  to  the 
triangle  ABC).  Hence  the  rectangle  EFNP  is  nine 
times  the  rectangle  BCLM,  and,  therefore,  the  triangle 
DEF  is  nine  times  the  triangle  ABC. 
That  is,  when 

side  BC:  side  EF  =  1:3, 
then,  triangle  ABC:  triangle  DEF  =  1:3% 
the  triangles  being,  of  course,  similar. 

140 


SiMILAE  TkIANGLES. 


141 


2.  Again,  let  ABC  and  DEF  be  similar  triangles, 
having  the  base  EF  one  and  three-quarter  times  the 
base  BC.  That  is,  the  base  BC  is  to  the  base  EF  as  4 
is  to  7,  since  1:1J  =  4:7.  Since  the  angles  are  similar, 
the  other  sides  of  DEF  are  IJ  times  the  correspond- 
ing sides  of  ABC.  If  AG  and  DK  be  the  perpendiculars 
to  the  bases,  the  triangles  ABG  and  DEK  are  equi- 
angular, and,  therefore,  since  DE  is  IJ  times  AB,  DK 
is  also  If  times  AG. 

If  rectangles  be  con- 
structed on  the  bases 
equal  to  the  triangles, 
the  heights  of  these 
rectangles  are  half  the 
heights  of  the  triangles 
(Ch.  VIII.,  5).  Hence 
FN,  which  is  half  of 
DK,  is  13  times  CL, 
which  is  half  of  AG. 

So  that  the  rectangle 
EFNP  (which  is  equal  to  the  triangle  DEF)  is  IJ  times 
as  long  and  IJ  times  as  high  as  the  rectangle  BCLM 
(which  is  equal  to  the  triangle  ABC).  That  is,  of  such 
parts  as  EF  contains  7,  BC  contains  4;  and  of  such 
parts  as  FN  contains  7,  CL  contains  4.  Hence  of  such 
small  areas  as  the  rectangle  EFNP  contains  7^=49, 
the  rectangle  BCLM  contains  4^  =  16.  And  therefore 
the  triangle  ABC  is  to  the  triangle  DEF  as  16  is  to  49. 

That  is,  when 

BC:EF  =  l;lf  =  4:7, 

then,  triangle  ABC:  triangle  DEF  =  16  :  49  =  4^  :  7^ 

or  1:(1J)^ 


i. 


K 


142  Geometry. 

3.  Make  figures  as  in  §  1  and  §  2  for  the  following 
problems : 

Two  similar  triangles,  ABC  and  DEF,  have  their  cor- 
responding sides  BC  and  EF,  1  and  2  inches  in  length 
respectively ;  show  that  their  areas  are  as  1  to  4,  i.e.^ 
as  1  to  2K 

Two  similar  triangles,  ABC  and  DEF,  have  their  cor- 
responding sides  BC  and  EF,  1  and  1 J  inches  in  length 
respectively ;  show  that  their  areas  are  as  4  to  9,  i.e., 
as  1  to  (1J)2. 

Two  similar  triangles,  ABC  and  DEF,  have  their  cor- 
responding sides  BC  and  EF,  30  and  50  millimetres  in 
length  respectively ;  show  that  their  areas  are  as  9  to 
25,  i.e.,  as  (30)^  to  (50)^ 

(For  the  three  preceding  constructions,  the  method 
of  article  4,  which  follows,  should  also  be  employed.) 

The  result  of  our  observations  in  such  cases  as  the 
preceding  may  be  stated  thus: 

Similar  triangles  are  to  one  another  as  the 
squares  of  corresponding  sides. 

Note:  In  the  preceding  examples  it  will  be  ob- 
served that  the  lengths  of  the  corresponding  sides  are 
supposed  commensurable,  i.e.,  a  unit  of  length  can  be 
found  that  is  contained  in  both  an  exact  number  of 
times.  All  lines  are  not  commensurable,  though  the 
preceding  statement  in  italics  is  true  of  all  similar 
triangles,  whether  the  corresponding  sides  be  commen- 
surable or  not. 

4.  The  following  is  possibly  a  more  striking  way  of 
presenting  the  preceding  proposition: 


Similar  Teiangles.  143 

Let  any  side,  say  the  base,  of  a  triangle  be  divided 
into  as  many  parts  as  it  contains  units  of  length. 
Through  the  points   of   division  draw  lines  parallel  to 


the  sides,  and,  through  the  points  of  intersection  of 
these  lines,  draw  lines  parallel  to  the  base.  The  tri- 
angle is  thus  divided  into  a  number  of  triangles  equal 
to  one  another  in  all  respects,  and  all  similar  to  the 
original  triangle.  It  will  be  observed  that,  considering 
these  triangles  in  rows,  the  rows  contain  1,  3,  5,  7,  .  . 
triangles,  respectively.  Hence  if  the  base  be  2  units 
in  length,  the  large  triangle  contains  1  +  3  =  2^  small 
triangles;  if  3  units  in  length,  1  +  3  +  5  =  3^  small 
triangles;  if  4  units  in  length,  1  +  3  +  5  +  7  =  4^  small 
triangles;  and  so  on.  Thus  if  there  be  two  similar 
triangles,  the  base  of  one  containing  3  units  of  length, 
and  the  base  of  the  other  4  units  of  length,  the 
number  of  small  triangles  in  one  will  be  3^,  and  in 
the  other  4^,  all  such  triangles  being  equal  to  one 
another.  Hence  the  areas  of  the  triangles  are  as  3^ 
to  4-,  i.e.,  as  the  squares  of  the  bases. 


144  Geometry. 


Exercises.  . 

4.  Construct  two  angles,  the  sides  of  one  being  36,  48  and  50,  and 
the  sides  of  the  other  54,  72,  75  millimetres.  On  the  base  of  each 
construct  a  rectangle  equal  to  it ;  and  divide  up  the  rectangles  so 
as  to  show  that  the  triangles  are  as  (36)^  to  (54)^. 

2.  Divide  the  triangles  of  the  preceding  question  into  smaller 
triangles,  all  equal  to  one  another.  Hence  show  that  the  original 
triangles  are  as  (48)^  to  (72)  \ 

3.  Draw  two  straight  lines  which  are  to  one  another  as  these 
triangles. 

4.  Divide  a  line  3^  in.  in  length  into  two  segments,  such  that, 
when  equilateral  triangles  are  described  on  the  segments,  one 
triangle  shall  be  four  times  the  other. 

Construct  the  equilateral  triangles,  and  divide  the  greater  into 
four  triangles,  each  equal  to  the  smaller. 

5.  Construct  two  triangles  on  bases  of  45  and  75  millimetres,  with 
angles  adjacent  to  each  base  70°  and  50°.  Divide  the  triangles  into 
smaller  ones,  all  equal  to  one  another,  showing  that  the  areas  of  the 
triangles  are  as  (45)^  to  (75j^. 

6.  Draw  a  line  AB  of  length  1  in.,  and  produce  it  to  C  so  that  AB 
may  be  to  BC  as  the  areas  of  the  two  triangles  in  the  preceding 
question. 

7.  Describe  an  irregular  pentagon,  and,  after  the  manner  of  §  5, 
Ch.  XX.,  construct  another  pentagon  with  linear  dimensions  half 
those  of  former.  Divide  each  pentagon  into  three  triangles  by  lines 
drawn  from  corresponding  angles. 

How  are  the  sides  and  angles  of  corresponding  triangles  related  ? 
Test  with  bevel  and  dividers. 

How  many  times  is  a  triangle  in  the  first  pentagon  greater  than 
the  corresponding  triangle  in  the  second  ? 

How  many  times  is  one  pentagon  greater  than  the  other  ? 

8.  ABC  is  any  triangle,  and  in  AB  a  point  D  is  taken  such  that 
AD  is  one-quarter  of  AB.  DE  is  drawn  parallel  to  BC.  What  frac- 
tional part  is  ADE  of  the  whole  triangle  ?  What  ratio  does  ADE 
bear  to  the  rest  of  ABC  ? 


Exercises.  145 

9.  Construct  an  equilateral  triangle  with  sides  of  H  in-r  and  con- 
struct another  with  area  twice  the  former. 

10.  Construct  a  right-angled  triangle  with  sides  30,  40  and  50  mil- 
limetres. On  the  sides  describe  equilateral  triangles.  Divide  the 
triangles  into  smaller  ones,  so  that  the  smaller  ones  may  all  be  equal 
to  one  another.  What  relation  do  you  discover  between  the  area  of 
the  triangle  on  the  hypotenuse  and  the  areas  of  the  two  other 
triangles  ? 

11.  In  the  preceding  question,  instead  of  equilateral  triangles,  con- 
struct triangles  with  angles  adjacent  to  the  sides  of  50°  and  80°,  so 
that  the  three  triangles  are  similar.  Again  compare  areas  of  smaller 
triangles  with  area  of  greatest. 

12.  Any  line  being  taken  as  unity,  construct  for  other  lines  which 
shall  represent  v/2  and  ^. 

Hence  draw  lines  parallel  to  the  base  of  any  triangle  so  as  to 
form  with  sides,  or  sides  produced,  triangles  half  and  twice  the 
original. 

13.  The  areas  of  the  following  states  being,  —  Texas,  265780 ;  New 
York,  49170;  Illinois,  56650;  California,  158360;  Kansas,  82080; 
Massachusetts,  8315 ;  South  Carolina,  30570  square  miles ;  and  the 
square  roots  of  these  numbers  being  515,  222,  238,  398,  286,  91,  175,  or 
approximately  as  52,  22,  24,  40,  29,  9,  18 ;  construct  seven  equilateral 
triangles,  all  with  the  same  vertex,  whose  areas  shall' represent  pro- 
portionately the  areas  of  these  states. 

14.  Draw  also  seven  parallel  Ihies,  near  one  another,  and  all 
terminated  at  one  end  by  the  same  straight  line  to  which  they  are 
perpendicular,  so  that  these  lines  may  approximately  represent  the 
areas  of  these  states. 

15.  Given  the  following  populations, — Pennsylvania,  6302115 ;  Ohio, 
4157545;  Missouri,  3106665;  Indiana,  2516462;  Vermont,  343641; 
construct  five  squares,  with  one  angle  in  common,  which  shall  repre- 
sent proportionately  and  approximately  the  populations  of  these  states. 

16.  Draw  also  five  parallel  lines,  as  in  14,  which  shall  represent 
approximately  the  populations  of  these  states. 


146  Geometry. 

17.  ABC  is  a  right-angled  triangle  (C  =  90°),  and  CD  is  drawn 
perpendicular  to  AB. 

(i)   Prove  that  triangles  CAD  and  BAC  are  equiangular; 

also  CBD  and  ABC  equiangular, 
(2)  Hence  show  that 

AC2  +  BC2  _ 
AB2         ~  ^' 

18.  The  school  expenditures  for  certain  cities,  in  1901-2,  being  as 
follows:  San  Francisco,  $1331541;  St.  Louis,  11636575;  Boston, 
$4007264;  Philadelphia,  $4223277;  Chicago,  $8511019;  New  York, 
$23013600 ;  construct  equilateral  triangles,  with  one  angle  common, 
whose  areas  shall  proportionately  and  approximately  represent  these 
expenditures,  the  side  of  the  first  triangle  being  12  millimetres. 

19.  What  will  be  the  sides  of  the  triangles  if  their  perimeters  are 
to  represent  the  expenditures,  the  side  of  the  first  being  again  12  milli- 
metres ? 

20.  Construct  two  triangles,  the  sides  of  one  being  twice  the  sides 
of  the  other,  and  ascertain  the  following: 

(1)  The  ratio  of  perpendiculars  from  corresponding  angles  on 

opposite  sides. 

(2)  The  ratio  of  corresponding  segments  (of  sides)  made  by 

feet  of  perpendiculars. 

(3)  The  ratio  of  lines  from  corresponding  angles  to  bisections 

of  opposite  sides. 


14  DAY  USE 

RETURN  TO  DESK  FROM  WHICH  BORROWED 

LOAN  DEPT. 

This  book  is  due  on  the  last  date  stamped  below,  or 

on  the  date  to  which  renewed. 

Renewed  books  are  subject  to  immediate  recall. 

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